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A318696
Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).
6
1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392
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 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*tau(d) ) * x^k/k).
MAPLE
seq(n!*coeff(series(mul((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 + 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 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) 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[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]/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] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2018 *)
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 31 2018
STATUS
approved