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A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).

%I A321387 #15 Sep 08 2022 08:46:23
%S A321387 1,1,2,11,74,708,8583,127424,2239965,45514345,1049365071,27061132159,
%T A321387 771695223819,24109698083919,818914886275467,30044684789498522,
%U A321387 1184048086192376822,49883929845112421452,2237287911899357657492,106426388125032988691636,5352033610656721914626572
%N A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).
%C A321387 Weigh transform of A000169.
%H A321387 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A321387 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^d ) * x^k/k).
%F A321387 a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-1)/2 + 2*exp(-2))/n^2). - _Vaclav Kotesovec_, Nov 09 2018
%p A321387 a:=series(mul((1+x^k)^(k^(k-1)),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # _Paolo P. Lava_, Apr 02 2019
%t A321387 nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A321387 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
%o A321387 (PARI) seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^d ) * x^k/k) + O(x*x^n)))} \\ _Andrew Howroyd_, Nov 09 2018
%o A321387 (PARI) m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-1)))) \\ _G. C. Greubel_, Nov 09 2018
%o A321387 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^(k-1)): k in [1..m]]) )); // _G. C. Greubel_, Nov 09 2018
%Y A321387 Cf. A000169, A023879, A261053, A283335, A321385, A321388.
%K A321387 nonn
%O A321387 0,3
%A A321387 _Ilya Gutkovskiy_, Nov 08 2018