A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k)).

1, 1, 3, 6, 12, 21, 43, 70, 127, 215, 364, 591, 989, 1562, 2515, 3954, 6194, 9538, 14754, 22349, 33926, 50910, 76102, 112721, 166747, 244205, 356984, 518344, 749924, 1078711, 1547668, 2207418, 3140135, 4446572, 6276657, 8823776, 12371487, 17275879, 24061878

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. A007425, A107742, A174465, A219560, A280486.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A007425[[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, Aug 30 2018 *)

Formula

G.f.: Product_{k>=1} (1 + x^k)^tau_3(k), where tau_3() = A007425. - Ilya Gutkovskiy, May 22 2018