A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l)).

1, 1, 4, 8, 20, 36, 86, 150, 314, 564, 1088, 1902, 3557, 6085, 10902, 18506, 32124, 53584, 91133, 149749, 249315, 405121, 662582, 1063152, 1714580, 2719842, 4327302, 6797316, 10686005, 16622003, 25861855, 39866017, 61422891, 93910783, 143406552, 217537696

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. A000009, A007426, A107742, A280473, A280487, A219561.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(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]*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}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Sep 08 2018 *)

Formula

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