A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(i*j*k*l)).

1, 1, 5, 9, 29, 49, 135, 235, 565, 995, 2177, 3821, 7900, 13728, 26974, 46606, 88128, 150644, 276283, 467647, 835708, 1400874, 2448818, 4065230, 6975307, 11470265, 19359345, 31552473, 52488142, 84808548, 139274675, 223191639, 362297234, 576064732, 925295844

Offset: 0

Author

Vaclav Kotesovec, Jan 04 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• 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. A000041, A006171, A007426, A174465, A280486.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[tau4[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2018 *)

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018