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A318695
Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).
5
1, 1, 4, 16, 106, 658, 6088, 51952, 592828, 6577948, 88213744, 1173121024, 18663391096, 289030343704, 5157010548064, 92428084599232, 1848308567352592, 37038307949425168, 822602470902709312, 18285742807660340992, 444405771941314880416, 10883864256927386369056, 286778106663948874858624
OFFSET
0,3
LINKS
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}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 31 2018
STATUS
approved