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A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

%I A261053 #18 Sep 08 2022 08:46:13
%S A261053 1,1,4,31,289,3495,51268,891152,17926913,409907600,10499834497,
%T A261053 297793199060,9262502810645,313457634240463,11464902463397642,
%U A261053 450646709610954343,18943070964019019671,847932498252050293971,40266255926484893366914,2021845081107882645459639
%N A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).
%H A261053 Alois P. Heinz, <a href="/A261053/b261053.txt">Table of n, a(n) for n = 0..385</a>
%F A261053 a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2).
%F A261053 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d+1) ) * x^k/k). - _Ilya Gutkovskiy_, Nov 08 2018
%p A261053 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261053       add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
%p A261053     end:
%p A261053 a:= n-> b(n$2):
%p A261053 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 08 2015
%t A261053 nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]
%o A261053 (PARI) m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ _G. C. Greubel_, Nov 08 2018
%o A261053 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // _G. C. Greubel_, Nov 08 2018
%Y A261053 Cf. A023880, A261052, A026007, A027998, A248882, A102866, A256142.
%K A261053 nonn
%O A261053 0,3
%A A261053 _Vaclav Kotesovec_, Aug 08 2015