A000785 -id:A000785

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page 1

A259096

a(n) = A000785(n) - A005575(n-1).

+20
0

0, 0, 0, 0, 0, 0, 1, 2, 10, 19, 52, 105, 224, 429, 820, 1484, 2668, 4627, 7928, 13305, 22050, 35988, 58128, 92792, 146790, 230082, 357831, 552310, 846876, 1290231, 1954453, 2944513, 4413897, 6585072, 9780996, 14466926, 21313816, 31283424, 45753928, 66691357, 96897846

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,8

COMMENTS

Proposed and computed by R. K. Guy in 1988.

LINKS

Table of n, a(n) for n=2..42.

R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission).

CROSSREFS

Cf. A000785, A005575.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 20 2015

EXTENSIONS

More terms from Alois P. Heinz, Jul 11 2016

More terms from Amiram Eldar, May 23 2024

STATUS

approved

A000219

Number of planar partitions (or plane partitions) of n.
(Formerly M2566 N1016)

+10
273

1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, 8512309, 12733429, 18974973, 28175955, 41691046, 61484961, 90379784, 132441995, 193487501, 281846923

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Two-dimensional partitions of n in which no row or column is longer than the one before it (compare A001970). E.g., a(4) = 13:

4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2

.....1....2.....1...1......1...11.1..1........ 11

....................1.............1..1

.....................................1

In the above, one also must require that rows & columns are nondecreasing, e.g., [1,1; 2] is also forbidden (which implies that row and column lengths are nondecreasing, if empty cells are identified with cells filled with 0's). - M. F. Hasler, Sep 22 2018

Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner. - Wouter Meeussen

Also number of partitions of n objects of 2 colors, each part containing at least one black object; see example. - Christian G. Bower, Jan 08 2004

Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. E.g., n=3 gives 111, 12, 12', 3, 3', 3''. - Jon Perry, May 27 2004

The bijection between the partitions in the two preceding comments goes by identifying a part with k black objects with a part of type k. - David Scambler and Joerg Arndt, May 01 2013

Can also be regarded as the number of Jordan canonical forms for an n X n matrix. (I.e., a 5 X 5 matrix has 24 distinct Jordan canonical forms, dependent on the algebraic and geometric multiplicity of each eigenvalue.) - Aaron Gable (agable(AT)hmc.edu), May 26 2009

(1/n) * convolution product of n terms * A001157 (sum of squares of divisors of n): (1, 5, 10, 21, 26, 50, 50, 85, ...) = a(n). As shown by [Bressoud, p. 12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 48. - Gary W. Adamson, Jun 13 2009

Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13, ...) = A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83, ...). - Gary W. Adamson, Jun 13 2009

Starting with offset 1 = row sums of triangle A162453. - Gary W. Adamson, Jul 03 2009

Unfortunately, Wright's formula is also incomplete in the paper by G. Almkvist: "Asymptotic formulas and generalized Dedekind sums", p. 344, (the denominator should have sqrt(3*Pi) not sqrt(Pi)). This error was already corrected in the paper by Steven Finch: "Integer Partitions". - Vaclav Kotesovec, Aug 17 2015

Also the number of non-isomorphic weight-n chains of multisets whose dual is also a chain of multisets. The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Sep 25 2018

REFERENCES

G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.

L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Royal Soc., 211 (1912), 345-373.

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.

P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - N. J. A. Sloane, May 21 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Suresh Govindarajan, Table of n, a(n) for n = 0..6500 (first 401 terms from T. D. Noe)

G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

G. E. Andrews and P. Paule, MacMahon's partition analysis XII: Plane Partitions, J. Lond. Math. Soc., 76 (2007), 647-666.

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 18, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.

Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.

E. A. Bender and D. E. Knuth, Enumeration of Plane Partitions, J. Combin. Theory A. 13, 40-54, 1972.

S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, p. 41-42.

Henry Bottomley, Illustration of initial terms

D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.

Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016.

Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.

Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580.

Bernhard Heim, Markus Neuhauser and Robert Tröger, Inequalities for Plane Partitions, arXiv:2109.15145 [math.CO], 2021.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 18.

Vaclav Kotesovec, Graphs - The asymptotic ratio (250000 terms)

D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.

Oleg Lazarev, Matt Mizuhara and Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions, 13 August 2010.

P. A. MacMahon, Combinatory analysis.

J. Mangual, McMahon's Formula via Free Fermions, arXiv preprint arXiv:1210.7109 [math.CO], 2012. - From N. J. A. Sloane, Jan 01 2013

Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ..., arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.

L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions..., arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.

Ken Ono, Sudhir Pujahari and Larry Rolen, Turán inequalities for the plane partition function, arXiv:2201.01352 [math.NT], 2022.

I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006) 5-75.

A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.

Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.

N. J. A. Sloane, Transforms

J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geometry & Topology 12.2 (2008): 1171-1202.

Eric Weisstein's World of Mathematics, Plane Partition

E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.

Index entries for "core" sequences

FORMULA

G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.

Euler transform of sequence [1, 2, 3, ...].

a(n) ~ (c_2 / n^(25/36)) * exp( c_1 * n^(2/3) ), where c_1 = A249387 = 2.00945... and c_2 = A249386 = 0.23151... - Wright, 1931. Corrected Jun 01 2010 by Rod Canfield - see Mutafchiev and Kamenov. The exact value of c_2 is e^(2c)*2^(-11/36)*zeta(3)^(7/36)*(3*Pi)^(-1/2), where c = Integral_{y=0..inf} (y*log(y)/(e^(2*Pi*y)-1))dy = (1/2)*zeta'(-1).

The exact value of c_1 is 3*2^(-2/3)*Zeta(3)^(1/3) = 2.0094456608770137530649... - Vaclav Kotesovec, Sep 14 2014

a(n) = (1/n) * Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n) = A001157(n) = sum of squares of divisors of n. - Vladeta Jovovic, Jan 20 2002

G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Jan 10 2003

From Vaclav Kotesovec, Nov 07 2016: (Start)

More precise asymptotics: a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36) * n^(25/36))

* (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where

c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))

c2 = -0.02576771365117401620018082... = 353*Zeta(3)^(1/3)/(248832*2^(2/3)) - 17*Zeta(3)^(4/3)/(3225600*2^(2/3)) - 71575/(1492992*(2*Zeta(3))^(2/3))

c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000

and A = A074962 is the Glaisher-Kinkelin constant.

(End)

EXAMPLE

A planar partition of 13:

4 3 1 1

2 1

a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+ 15*sigma_2(1)*sigma_2(2)^2+30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) = 24. - Vladeta Jovovic, Jan 10 2003

From David Scambler and Joerg Arndt, May 01 2013: (Start)

There are a(4) = 13 partitions of 4 objects of 2 colors ('b' and 'w'), each part containing at least one black object:

1 black part:

[ bwww ]

2 black parts:

[ bbww ]

[ bww, b ]

[ bw, bw ]

3 black parts:

[ bbbw ]

[ bbw, b ]

[ bb, bw ]

(but not: [bw, bb ] )

[ bw, b, b ]

4 black parts:

[ bbbb ]

[ bbb, b ]

[ bb, bb ]

[ bb, b, b ]

[ b, b, b, b ]

(End)

The corresponding partitions of the integer 4 are:

4'''

4''

3'' + 1

2' + 2'

3' + 1

2 + 2'

2' + 1 + 1

3 + 1

2 + 2

2 + 1 + 1

1 + 1 + 1 + 1. - Geoffrey Critzer, Nov 29 2014

From Gus Wiseman, Sep 25 2018: (Start)

Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:

{{1},{1,1,1}}

{{2},{1,2,2}}

{{3},{1,2,3}}

{{1,1},{1,1}}

{{1,2},{1,2}}

{{1},{1},{1,1}}

{{2},{2},{1,2}}

{{1},{1},{1},{1}}

(End)

G.f. = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 48*x^6 + 86*x^7 + 160*x^8 + ...

MAPLE

series(mul((1-x^k)^(-k), k=1..64), x, 63);

# second Maple program:

a:= proc(n) option remember; `if`(n=0, 1, add(

a(n-j)*numtheory[sigma][2](j), j=1..n)/n)

end:

seq(a(n), n=0..50); # Alois P. Heinz, Aug 17 2015

MATHEMATICA

CoefficientList[Series[Product[(1 - x^k)^-k, {k, 64}], {x, 0, 64}], x]

Zeta[3]^(7/36)/2^(11/36)/Sqrt[3 Pi]/Glaisher E^(3 Zeta[3]^(1/3) (n/2)^(2/3) + 1/12)/n^(25/36) (* asymptotic formula after Wright; Vaclav Kotesovec, Jun 23 2014 *)

a[0] = 1; a[n_] := a[n] = Sum[a[n - j] DivisorSigma[2, j], {j, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)

CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* Eric W. Weisstein, Feb 01 2018 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, x^k / (1 - x^k)^2 / k, x * O(x^n))), n))}; /* Michael Somos, Jan 29 2005 */

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-k), n))}; /* Michael Somos, Jan 29 2005 */

(PARI) my(N=66, x='x+O('x^N)); Vec( prod(n=1, N, (1-x^n)^-n) ) \\ Joerg Arndt, Mar 25 2014

(PARI) A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - M. F. Hasler, Sep 24 2018

(Python)

from sympy import cacheit

from sympy.ntheory import divisor_sigma

@cacheit

def A000219(n):

if n <= 1:

return 1

return sum(A000219(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n

print([A000219(n) for n in range(20)])

# R. J. Mathar, Oct 18 2009

(Julia)

using Nemo, Memoize

@memoize function a(n)

if n == 0 return 1 end

s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)

return div(s, n)

end

[a(n) for n in 0:20] # Peter Luschny, May 03 2020

(SageMath) # uses[EulerTransform from A166861]

b = EulerTransform(lambda n: n)

print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

CROSSREFS

Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.

Cf. A023871-A023878, A026007, A001157, A162453, A285216.

Differences: A191659, A191660, A191661.

Row sums of A089353 and A091438 and A091298.

Column k=1 of A144048. - Alois P. Heinz, Nov 02 2012

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

Cf. A249386, A249387.

Cf. A161870, A255610, A255611, A255612, A255613, A255614, A193427.

KEYWORD

nonn,nice,easy,core

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected by N. J. A. Sloane, Jul 29 2006

Minor edits by Vaclav Kotesovec, Oct 27 2014

STATUS

approved

A048141

Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.

+10
13

1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 4, 0, 4, 5, 0, 5, 7, 1, 6, 9, 1, 6, 11, 1, 8, 15, 2, 10, 20, 3, 10, 25, 4, 12, 33, 7, 14, 40, 9, 15, 48, 12, 18, 60, 17, 20, 74, 23, 22, 89, 30, 26, 108, 40, 30, 130, 51, 33, 157, 66, 37, 187, 85, 42, 222, 108, 47, 262, 136, 54

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

LINKS

Wouter Meeussen, Table of n, a(n) for n=1..151

EXAMPLE

The plane partition {{2,1},{1}} has C3v symmetry.

CROSSREFS

Cf. A000219, A000784, A000785, A000786, A005987, A048142, A051056-A051061, A096419.

KEYWORD

nice,nonn

AUTHOR

Wouter Meeussen

STATUS

approved

A048142

Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.

+10
13

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 2, 2, 0, 3, 3, 0, 5, 6, 0, 7, 9, 0, 11, 16, 1, 14, 23, 2, 20, 36, 4, 27, 52, 7, 37, 78, 13, 48, 111, 21, 65, 163, 36, 83, 227, 56, 109, 322, 89, 139, 444, 135, 179, 618, 207, 226, 841, 305, 288, 1151, 453, 361

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,19

LINKS

Wouter Meeussen, Table of n, a(n) for n = 1..151

EXAMPLE

The plane partitions {{3, 2, 2}, {3, 1}, {1, 1}} and {{3, 2, 2}, {3, 2}, {1, 1}} have C3 symmetry.

CROSSREFS

Cf. A000219, A000784, A000785, A000786, A005987, A048141, A051056-A051061, A096419.

KEYWORD

nice,nonn

AUTHOR

Wouter Meeussen

STATUS

approved

A005987

Number of symmetric plane partitions of n.
(Formerly M0562)

+10
12

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 41, 53, 71, 93, 125, 160, 211, 270, 354, 450, 581, 735, 948, 1191, 1517, 1902, 2414, 3008, 3791, 4709, 5909, 7311, 9119, 11246, 13981, 17178, 21249, 26039, 32105, 39213, 48159, 58669, 71831, 87269

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

From M. F. Hasler, Sep 26 2018: (Start)

A plane partition of n is a matrix of nonnegative integers that sum up to n, and such that A[i,j] >= A[i+1,j], A[i,j] >= A[i,j+1] for all i,j. We can consider A of infinite size but there are at most n nonzero rows and columns and we ignore empty rows or columns. It is symmetric iff A = transpose(A), i.e., A[i,j] = A[j,i] for all i,j.

For any n, we have A000219(n) = a(n) + 2*A306098(n) where A306098(n) is the number of equivalence classes, modulo transposition, of non-symmetric plane partitions. (For any of these, its transpose is a different plane partition of n.) (End)

REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 134.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.20.5

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

A. Björner and R. P. Stanley, with A combinatorial miscellany, L'Enseignement Math., Monograph No. 42, 2010.

R. P. Stanley, Theory and application of plane partitions II, Studies in Appl. Math., 50 (1971), 259-279. DOI:10.1002/sapm1971503259. [Scan on author's personal web page].

FORMULA

G.f.: Product_{i=1..oo} 1/(1-x^(2i-1))/(1-x^(2i))^floor(i/2). (Stanley 1971, Prop.14.3; Björner & Stanley 2010, p. 33).

a(n) ~ exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * Zeta(3)^(1/3)) - Pi^4 / (384*Zeta(3)) + 1/24) * Zeta(3)^(13/72) / (2^(77/72) * sqrt(3*Pi*A) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 05 2018

EXAMPLE

From M. F. Hasler, Sep 26 2018: (Start)

The only plane partition of n = 0 is the empty partition []; we consider it to be symmetric (as a 0 X 0 matrix), so a(0) = 1.

The only plane partition of n = 1 is the partition [1] which is symmetric, so a(1) = 1.

For n = 2 we have the partitions [2], [1 1] and [1; 1] (where ; denotes the end of a row). Only the first one is symmetric, so a(2) = 1.

For n = 3 we have the partitions [3], [2 1], [2; 1], [1 1; 1 0], [1 1 1], [1; 1; 1]. The first and the fourth are symmetric, so a(3) = 2. (End)

MATHEMATICA

terms = 46; s = Product[1/(1 - x^(2i-1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[terms/2]}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 10 2017 *)

PROG

(PARI) a(n)=polcoeff(prod(k=1, n, (1-x^k)^-if(k%2, 1, k\4), 1+x*O(x^n)), n) \\ Michael Somos, May 19 2000

(PARI) show(n)=select(t->(t=matconcat(t~))~==t, PlanePartitions(n)) \\ Using PlanePartitions() given in A091298, this selects and returns the list of symmetric plane partitions of n. - M. F. Hasler, Sep 26 2018

CROSSREFS

Cf. A000784, A000785, A000786, A000219, A048142.

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wouter Meeussen, Dec 11 1999

Edited by M. F. Hasler, Sep 26 2018

STATUS

approved

A000786

Number of inequivalent planar partitions of n, when considering them as 3D objects.
(Formerly M1020 N0383)

+10
11

1, 1, 1, 2, 4, 6, 11, 19, 33, 55, 95, 158, 267, 442, 731, 1193, 1947, 3137, 5039, 8026, 12726, 20024, 31373, 48835, 75673, 116606, 178889, 273061, 415086, 628115, 946723, 1421082, 2125207, 3166152, 4700564, 6954151, 10254486, 15071903

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Partitions that are the same when regarded as 3-D objects are counted only once. - Wouter Meeussen, May 2006

REFERENCES

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..150

P. A. MacMahon, Combinatory analysis.

Eric Weisstein's World of Mathematics, Macdonald's Plane Partition Conjecture.

Eric Weisstein's World of Mathematics, Plane Partition.

FORMULA

Equals A000784 + A000785 + A048141 + A048142.

Equals (A048141 + 3*A048140 - A000219 + 2*A048142)/3. - Wouter Meeussen, May 2006

EXAMPLE

From M. F. Hasler, Oct 01 2018: (Start)

For n = 2, all three plane partitions [2], [1 1] and [1; 1] (where ";" means next row) correspond to a 1 X 1 X 2 rectangular cuboid, therefore a(2) = 1.

For n = 3, we have [3] ~ [1 1 1] ~ [1; 1; 1] all corresponding to a 1 X 1 X 3 rectangular cuboid or tower of height 3, and [2 1] ~ [2; 1] ~ [1 1; 1] correspond to an L-shaped object, therefore a(3) = 2.

For n = 4, [4] ~ [1 1 1 1] ~ [1; 1; 1; 1] correspond to the 4-tower; [3 1] ~ [3; 1] ~ [2 1 1] ~ [2; 1; 1] ~ [1 1 1; 1] ~ [1 1; 1; 1] all correspond to the same L-shaped object, [2 2] ~ [2; 2] ~ [1 1; 1 1] represent a "flat" square, and it remains [2, 1; 1], so a(4) = 4.

For n = 5, we again have the tower [5] ~ [1 1 1 1 1] ~ [1; 1; 1; 1; 1], a "narrow L" or 4-tower with one "foot" [4 1] ~ [4; 1] ~ [2 1 1 1] ~ [2; 1; 1; 1] ~ [1 1 1 1; 1] ~ [1 1; 1; 1; 1], a symmetric L-shape [3 1 1] ~ [3; 1; 1] ~ [1 1 1; 1; 1], a 3-tower with 2 feet [3 1; 1] ~ [2 1; 1; 1] ~ [2 1 1; 1], a flat 2+3 shape [3 2] ~ [3; 2] ~ [2 2 1] ~ [2; 2; 1] ~ [1 1 1; 1 1] ~ [1 1; 1 1; 1] and a 2X2 square with a cube on top, [2 1;1 1] ~ [2 2; 1] ~ [2 1; 2]. This yields a(5) = 6 classes. (End)

MATHEMATICA

nmax = 150;

a219[0] = 1;

a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;

s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[( nmax+1)/2]}] + O[x]^(nmax+1);

A005987 = CoefficientList[s, x];

a048140[n_] := (a219[n] + A005987[[n+1]])/2;

A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];

A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {_, _}][[All, 2]];

a[0] = 1;

a[n_] := (A048141[[n]] + 3 a048140[n] - a219[n] + 2 A048142[[n]])/3;

a /@ Range[0, nmax] (* Jean-François Alcover, Dec 28 2019 *)

CROSSREFS

Cf. A000784, A000785, A000219, A005987, A048142, A051056-A051061, A096419.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wouter Meeussen, 1999

Name & links edited and a(0) = 1 added by M. F. Hasler, Sep 30 2018

STATUS

approved

A000784

Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane).
(Formerly M0322 N0119)

+10
9

0, 1, 2, 2, 4, 6, 6, 11, 16, 20, 28, 41, 51, 70, 93, 122, 158, 211, 266, 350, 450, 577, 730, 948, 1186, 1510, 1901, 2408, 2999, 3790, 4703, 5898, 7310, 9111, 11231, 13979, 17168, 21229, 26036, 32095, 39188, 48155, 58657, 71798, 87262, 106472, 129014

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

REFERENCES

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Jean-François Alcover, Table of n, a(n) for n = 1..150

P. A. MacMahon, Combinatory analysis.

MATHEMATICA

nmax = 150;

a219[0] = 1;

a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;

s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);

A005987 = CoefficientList[s, x];

a048140[n_] := (a219[n] + A005987[[n + 1]])/2;

A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];

a[1] = 0;

a[n_] := -A048141[[n]] + 2 a048140[n] - a219[n];

a /@ Range[1, nmax] (* Jean-François Alcover, Dec 28 2019 *)

CROSSREFS

Equals -A048141 + 2*A048140 - A000219.

Cf. A000785, A000786, A005987, A048142.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wouter Meeussen

STATUS

approved

A048140

Number of planar partitions of n, but partitions that are mirror images of each other (when regarded as 3-D objects) are counted only once.

+10
5

1, 2, 4, 8, 14, 27, 47, 86, 149, 261, 444, 760, 1269, 2119, 3486, 5711, 9247, 14906, 23800, 37816, 59622, 93528, 145759, 226071, 348612, 535131, 817280, 1242824, 1881310, 2836377, 4258509, 6369669, 9491142, 14092537, 20851146, 30749471

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Plane partitions seen as 3-dimensional-objects can have a mirror symmetry plane.

LINKS

Jean-François Alcover, Table of n, a(n) for n = 1..1000

EXAMPLE

n=3 gives 4 forms: {{3}}; {{1,1,1}}={{1},{1},{1}}; {{2,1}}={{2},{1}}; {{1,1},{1}}.

MATHEMATICA

terms = 100;

a219[0] = 1;

a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;

s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[ (terms+1)/2]}] + O[x]^(terms+1);

A005987 = CoefficientList[s, x];

a[n_] := (a219[n] + A005987[[n+1]])/2;

a /@ Range[terms] (* Jean-François Alcover, Dec 28 2019 *)

CROSSREFS

Equals (A000219+A005987)/2.

Equals 2 Cs + 3 C1 + C3 + C3v, Cs=A000784, C1=A000785, C3=A048142, C3v=A048141. Cf. A000219, A005987.

KEYWORD

nonn

AUTHOR

Wouter Meeussen

EXTENSIONS

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

STATUS

approved

A048139

Number of planar partitions of n, when partitions that are rotations of each other (when regarded as 3-D objects) are counted only once.

+10
1

1, 1, 2, 5, 8, 16, 30, 54, 94, 168, 287, 493, 831, 1391, 2293, 3769, 6114, 9867, 15782, 25098, 39598, 62165, 96935, 150398, 232021, 356261, 544220, 827758, 1253222, 1889655, 2837455, 4244505, 6324993, 9392009, 13897056, 20494991, 30126628

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Plane partitions seen as 3-dimensional-objects can have a threefold symmetry axis.

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

n=3 gives 2 forms: {{3}}={{1,1,1}}={{1},{1},{1}} and {{2,1}}={{1,1},{1}}={{2},{1}}.

CROSSREFS

Equals Cs + 2 C1 + 2 C3 + C3v, Cs=A000784, C1=A000785, C3=A048142, C3v=A048141. Cf. A000219, A005987.

Or, equals (2*A048141+A000219+4*A048142)/3.

KEYWORD

nonn

AUTHOR

Wouter Meeussen

EXTENSIONS

Edited by N. J. A. Sloane, Oct 26 2008 at the suggestion of R. J. Mathar.

STATUS

approved

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