A255835 -id:A255835

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

A253289

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

+10
10

1, 1, 4, 9, 22, 46, 103, 208, 431, 849, 1671, 3195, 6079, 11321, 20937, 38146, 68931, 123121, 218212, 383019, 667425, 1153544, 1980268, 3375394, 5717773, 9624541, 16108496, 26807662, 44379189, 73089219, 119789926, 195401275, 317309532, 513025167, 826000651

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

OFFSET

0,3

COMMENTS

a(n) is the number of partitions of n where there are 2*k-1 sorts of parts k. - Joerg Arndt, Aug 15 2020

LINKS

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

Vaclav Kotesovec, Graph - The asymptotic ratio

FORMULA

a(n) ~ 2^(1/9) * Zeta(3)^(1/18) * 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 * 3^(1/2) * n^(5/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

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

Euler transform of A005408 (the odd numbers). - Georg Fischer, Aug 15 2020

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); # after Alois P. Heinz

MATHEMATICA

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

(* Using EulerTransforms from 'Transforms'. *)

Prepend[EulerTransform[Table[2k + 1, {k, 0, 20}]], 1] (* Peter Luschny, Aug 15 2020 *)

CROSSREFS

Cf. A120844, A255802, A255835.

Cf. A005408.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 07 2015

STATUS

approved

A255834

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

+10
8

1, 3, 8, 23, 55, 129, 291, 627, 1317, 2697, 5398, 10589, 20421, 38743, 72452, 133724, 243792, 439496, 784070, 1385195, 2424971, 4209094, 7247141, 12383496, 21008559, 35398548, 59259781, 98595110, 163077878, 268221706, 438791204, 714142139, 1156552537

(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 / (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^(7/6)* 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

MATHEMATICA

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

CROSSREFS

Cf. A026007, A120844, A219555, A255835, A255836.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 07 2015

STATUS

approved

A261452

Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2*k-1).

+10
6

1, 2, 8, 24, 66, 176, 448, 1096, 2608, 6042, 13664, 30280, 65856, 140800, 296432, 615264, 1260306, 2550368, 5102616, 10101000, 19797344, 38439088, 73976160, 141179480, 267300752, 502283714, 937077808, 1736296304, 3196144032, 5846632656, 10631038400

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

OFFSET

0,2

COMMENTS

Convolution of A253289 and A255835.

LINKS

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 26.

FORMULA

a(n) ~ 2^(1/3) * (7*Zeta(3))^(1/18) * exp(1/6 - Pi^4/(672*Zeta(3)) - Pi^2 * n^(1/3)/(4*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * sqrt(3) * n^(5/9)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A156616, A261386, A005309, A261451, A261389.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Aug 19 2015

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

A261384

Expansion of Product_{k>=1} (1+x^k)^(2*k-1) / (1-x^k)^(2*k).

+10
3

1, 3, 12, 39, 117, 331, 893, 2307, 5766, 13986, 33046, 76302, 172567, 383013, 835731, 1795236, 3801105, 7941439, 16386777, 33423342, 67435311, 134675784, 266385932, 522135379, 1014643823, 1955656848, 3740191268, 7100290646, 13383997996, 25058666367

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

OFFSET

0,2

COMMENTS

Convolution of A161870 and A255835.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

FORMULA

a(n) ~ (7*Zeta(3))^(2/9) * exp(1/6 - Pi^4/(6048*Zeta(3)) - Pi^2 * n^(1/3) / (12*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * 2^(1/6) * sqrt(3*Pi) * n^(13/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k-1)/(1-x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A006950, A015128, A156616, A161870, A255835, A261386.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Aug 17 2015

STATUS

approved

A305655

a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^n)).

+10
1

1, 1, 3, 13, 54, 290, 1674, 10857, 76398, 580230, 4706734, 40598349, 370694845, 3569027696, 36100349833, 382360758863, 4228730647420, 48716663849192, 583403253712747, 7248883337962522, 93291181556742684, 1241632098163126324, 17064777292709034968, 241874821482784132204

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

N. J. A. Sloane, Transforms

FORMULA

a(n) = [x^n] Product_{k>=1} (1 + x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

MATHEMATICA

Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]

Table[SeriesCoefficient[Product[(1 + x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

CROSSREFS

Cf. A027998, A255835, A281156, A293554, A305206, A305654.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jun 07 2018

STATUS

approved

A363599

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

+10
1

1, 0, 3, 8, 18, 48, 109, 264, 594, 1360, 2988, 6552, 14115, 30048, 63288, 131800, 271953, 555792, 1126583, 2264472, 4518051, 8948544, 17603781, 34405272, 66828247, 129040704, 247765665, 473160696, 898924929, 1699331808, 3197083220, 5987288352, 11162934948

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..32.

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, A052812, A255835, A363600.

Cf. A363601.

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Jun 10 2023

EXTENSIONS

Name suggested by Joerg Arndt, Jun 11 2023

STATUS

approved

A319669

Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

+10
0

1, -1, -3, -2, 1, 10, 13, 15, -1, -30, -63, -89, -80, -14, 131, 304, 493, 561, 434, -32, -836, -1895, -2960, -3583, -3240, -1338, 2401, 8004, 14499, 20494, 23369, 20401, 8567, -13741, -46408, -85717, -124027, -149612, -147167, -101002, 2520, 168026, 388077, 634914

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..43.

FORMULA

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).

MAPLE

a:=series(mul((1-x^k)^(2*k-1), k=1..100), x=0, 44): seq(coeff(a, x, n), n=0..43); # Paolo P. Lava, Apr 02 2019

MATHEMATICA

nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

CROSSREFS

Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Sep 25 2018

STATUS

approved