A255834 -id:A255834

A219555

Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

+10
15

1, 2, 4, 10, 19, 38, 73, 134, 242, 430, 749, 1282, 2171, 3622, 5979, 9770, 15802, 25334, 40288, 63560, 99554, 154884, 239397, 367800, 561846, 853584, 1290107, 1940304, 2904447, 4328184, 6422164, 9489940, 13967783, 20480534, 29920277, 43557272, 63194864

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

OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)

FORMULA

a(n) = Sum_{i+j=n} [x^i*y^j] 1/2 * Product_{k,m>=0} (1+x^k*y^m).

G.f.: Product_{k>=1} (1+x^k)^(k+1). - Vaclav Kotesovec, Mar 07 2015

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (1296*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * 3^(4/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(2 - x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Aug 11 2018

EXAMPLE

a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))

end:

a:= n-> b(n$2):

seq(a(n), n=0..42); # Alois P. Heinz, Sep 19 2019

MATHEMATICA

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k+1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2015 *)

CROSSREFS

Row sums of A054242.

Cf. A026007, A052812, A005380, A255834, A255836.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 22 2012

STATUS

approved

A255835

G.f.: Product_{k>=1} (1+x^k)^(2*k-1).

+10
11

1, 1, 3, 8, 15, 34, 67, 133, 255, 486, 901, 1649, 2984, 5312, 9373, 16342, 28221, 48283, 81928, 137858, 230278, 381919, 629156, 1029933, 1675856, 2711288, 4362575, 6983196, 11122327, 17630798, 27820283, 43706461, 68375137, 106534093, 165340844, 255643289

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

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (2592*Zeta(3)) - Pi^2 * n^(1/3) / (12*(3*Zeta(3))^(1/3)) + 3^(4/3)/2 * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Jun 07 2018

MATHEMATICA

nmax=50; CoefficientList[Series[Product[(1+x^k)^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A026007, A219555, A052812, A253289, A255834.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 07 2015

STATUS

approved

A052812

A simple grammar: power set of pairs of sequences.

+10
10

1, 0, 1, 2, 3, 6, 9, 16, 24, 42, 63, 102, 157, 244, 373, 570, 858, 1290, 1930, 2858, 4228, 6208, 9084, 13216, 19175, 27666, 39804, 57020, 81412, 115820, 164264, 232178, 327220, 459796, 644232, 900214, 1254554, 1743896, 2418071, 3344896, 4616026

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

OFFSET

0,4

COMMENTS

Number of partitions of n objects of two colors into distinct parts, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Dec 28 2006

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 776

FORMULA

G.f.: exp(Sum((-1)^(j[1]+1)*(x^j[1])^2/(x^j[1]-1)^2/j[1], j[1]=1 .. infinity))

G.f.: Product_{k>=1} (1+x^k)^(k-1). - Vladeta Jovovic, Sep 17 2002

Weigh transform of b(n) = n-1. - Franklin T. Adams-Watters, Dec 28 2006

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4/(1296*Zeta(3)) - Pi^2 * n^(1/3) / (3^(4/3) * 2^(5/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/4) * 3^(1/3) * n^(2/3) * sqrt(Pi)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015

MAPLE

spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(B, B), S= PowerSet(C)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

MATHEMATICA

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2015 *)

CROSSREFS

Cf. A026007, A052847, A219555, A255834, A255835.

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from Vladeta Jovovic, Sep 17 2002

STATUS

approved

A120844

Number of multi-trace BPS operators for the quiver gauge theory of the orbifold C^2/Z_2.

+10
6

1, 3, 11, 32, 90, 231, 576, 1363, 3141, 7003, 15261, 32468, 67788, 138892, 280103, 556302, 1089991, 2108332, 4030649, 7620671, 14261450, 26431346, 48544170, 88393064, 159654022, 286149924, 509137464, 899603036, 1579014769

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

OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

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

Vaclav Kotesovec, Graph - The asymptotic ratio

FORMULA

G.f.: exp( Sum_{n>0} (3*x^n - x^(2*n)) / (n*(1-x^n)^2) ).

a(n) ~ Zeta(3)^(7/18) * exp(1/6 - Pi^4/(864*Zeta(3)) + Pi^2 * n^(1/3)/(3 * 2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 2^(2/9) * 3^(1/2) * Pi * n^(8/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015

MAPLE

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> 2*n+1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 06 2015 after Alois P. Heinz

MATHEMATICA

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

CROSSREFS

Cf. A005380, A253289, A255271, A255802, A255834.

KEYWORD

nonn

AUTHOR

Amihay Hanany (hanany(AT)mit.edu), Aug 25 2006

STATUS

approved

A255836

G.f.: Product_{k>=1} (1+x^k)^(3*k+1).

+10
5

1, 4, 13, 42, 117, 310, 785, 1896, 4433, 10062, 22248, 48080, 101821, 211682, 432795, 871520, 1730491, 3391894, 6568996, 12580316, 23841774, 44742634, 83193865, 153347110, 280336704, 508499474, 915540681, 1636805438, 2906642396, 5128530946, 8993376689

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

OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (3888*Zeta(3)) + Pi^2 * n^(1/3) / (6^(5/3) * Zeta(3)^(1/3)) + 3^(5/3)/2^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(17/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

MATHEMATICA

nmax=50; CoefficientList[Series[Product[(1+x^k)^(3*k+1), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A026007, A219555, A052812, A255271, A255834, A255835, A255837.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 07 2015

STATUS

approved

A255837

G.f.: Product_{k>=1} (1+x^k)^(3*k+2).

+10
3

1, 5, 18, 61, 182, 506, 1338, 3369, 8172, 19197, 43833, 97636, 212748, 454461, 953505, 1968095, 4001627, 8024295, 15885484, 31074351, 60111277, 115071431, 218126868, 409662895, 762679151, 1408172844, 2579599582, 4690277001, 8467363674, 15182486586

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

OFFSET

0,2

COMMENTS

In general, if g.f. = Product_{k>=1} (1+x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(1/6) * exp(-c^2 * Pi^4 / (1296*m*Zeta(3)) + (c * Pi^2 * n^(1/3)) / (2^(5/3) * 3^(4/3) * (m*Zeta(3))^(1/3)) + 3^(4/3) * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(4/3)) / (2^(m/12 + c/2 + 2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Mar 08 2015

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4/(972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(5/3) * Zeta(3)^(1/3)) + 3^(5/3)/2^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(23/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

MATHEMATICA

nmax=50; CoefficientList[Series[Product[(1+x^k)^(3*k+2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A026007 (k), A219555 (k+1), A052812 (k-1), A255834 (2*k+1), A255835 (2*k-1), A255836 (3*k+1).

Cf. A255803.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 07 2015

STATUS

approved

A363600

Number of partitions of n into distinct parts where there are k^2+1 kinds of part k.

+10
1

1, 2, 6, 20, 52, 140, 356, 880, 2123, 5016, 11610, 26400, 59130, 130476, 284216, 611592, 1301344, 2740194, 5713930, 11806144, 24184908, 49142504, 99091244, 198360536, 394342884, 778818658, 1528531702, 2982017956, 5784365082, 11158728448, 21413292868

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..30.

FORMULA

G.f.: Product_{k>=1} (1+x^k)^(k^2+1).

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2+1) ) * a(n-k).

PROG

(PARI) my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2+1)))

CROSSREFS

Cf. A027998, A219555, A255834, A363599.

Cf. A363602.

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Jun 10 2023

EXTENSIONS

Name suggested by Joerg Arndt, Jun 11 2023

STATUS

approved