A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

1, 1, 3, 7, 14, 28, 58, 106, 201, 372, 669, 1187, 2101, 3624, 6229, 10591, 17796, 29659, 49107, 80492, 131157, 212237, 341084, 544883, 865717, 1367233, 2148552, 3359490, 5227270, 8096544, 12486800, 19174319, 29326306, 44678825, 67811375, 102549673, 154545549

Offset: 0

Author

Joerg Arndt, Jun 24 2011

Keywords

• nonn

Comments

Euler transform of A002131. - Vaclav Kotesovec, Mar 26 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Crossrefs

Cf. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

Programs

Mathematica

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, Jul 03 2017 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

PARI

N=66;  x='x+O('x^N);
Q(x)=prod(k=1,N,1+x^k);
gf=prod(k=1,N, Q(x^k)^k );
Vec(gf) /* Joerg Arndt, Jun 24 2011 */

Formula

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 09 2017

a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018