A318696 - OEIS

%I #20 Sep 30 2024 21:10:02

%S 1,1,2,10,34,218,1708,12556,97340,1139932,12602584,142757624,

%T 1983086488,26745019000,402951386576,7181178238672,115410887636752,

%U 2039658743085584,42354537803172640,815690033731561888,17593347085888752416,416765224159172991136,9379433694333768563392

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

%H Vaclav Kotesovec, <a href="/A318696/b318696.txt">Table of n, a(n) for n = 0..446</a>

%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.

%F E.g.f.: Product_{k>=1} (1 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).

%F E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*tau(d) ) * x^k/k).

%p 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

%t 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]!

%t nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t 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]!

%t 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}]

%t 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 *)

%Y Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 31 2018