OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..440
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.
FORMULA
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau(k)/k), where tau = number of divisors (A000005).
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} tau(d) ) * x^k/k).
MAPLE
seq(n!*coeff(series(mul(1/(1-x^k)^(tau(k)/k), k=1..100), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019
MATHEMATICA
nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 31 2018
STATUS
approved