A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(1/(i*j*k)).

1, 1, 3, 15, 69, 477, 4167, 34731, 333225, 4058073, 48535659, 638782119, 9690930477, 146665611765, 2428164153711, 44904494549763, 820664075440593, 16238018609968689, 350155700132388435, 7568774583230565567, 175171222712837235861, 4318996957424273510541, 107317465474650443023383

Offset: 0

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..444
• 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. A000005, A007425, A168243, A280473, A318414, A318696, A318768, A318966.

Programs

Maple

a:=series(mul(mul(mul((1+x^(i*j*k))^(1/(i*j*k)),k=1..55),j=1..55),i=1..55),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[(-1)^(k/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(tau_3(k)/k), where tau_3 = A007425.

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1) * Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).