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%I #29 Feb 13 2024 07:01:14
%S 1,1,1,5,11,59,439,2659,13705,160649,2009681,16966421,183312931,
%T 2078169235,34203787591,657685416179,8054585463569,104530824746129,
%U 2595754682459425,39767021562661669,758079429084897211
%N Expansion of e.g.f. Product_{i>=1} (1 + x^i)^(1/i).
%H Vaclav Kotesovec, <a href="/A168243/b168243.txt">Table of n, a(n) for n = 0..446</a>
%H Vaclav Kotesovec, <a href="/A168243/a168243_2.jpg">Graph: (a(n)/n!) / (n^(log(2) - 1)), 250000 terms</a>
%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://arxiv.org/abs/2303.02240">A unified treatment of families of partition functions</a>, arXiv:2303.02240 [math.CO], 2023.
%F E.g.f.: exp(Sum_{n>=1} A048272(n)*x^n/n).
%F Conjecture: log(a(n)/n!) ~ (log(2) - 1) * log(n). - _Vaclav Kotesovec_, Sep 10 2018
%t nmax=20; CoefficientList[Series[Product[(1+x^k)^(1/k),{k,1,nmax}],{x,0,nmax}],x] * Range[0,nmax]! (* _Vaclav Kotesovec_, May 28 2015 *)
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 07 2018 *)
%Y Cf. A028342.
%K easy,nonn
%O 0,4
%A _Vladeta Jovovic_, Nov 21 2009
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