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%I #6 Apr 03 2019 02:55:14
%S 1,0,-1,-2,-2,-2,0,2,7,8,12,10,9,-2,-10,-32,-40,-62,-62,-70,-37,-20,
%T 57,106,224,272,388,376,431,272,192,-184,-414,-1012,-1321,-2020,-2157,
%U -2700,-2318,-2352,-1014,-272,2280,3798,7464,9200,13257,13958,17098,14846,15266
%N Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).
%C Convolution of A000009 and A255528.
%C Convolution inverse of A052812.
%F G.f.: exp(Sum_{k>=1} (-1)^k*x^(2*k)/(k*(1 - x^k)^2)).
%p a:=series(mul(1/(1+x^k)^(k-1),k=1..100),x=0,51): seq(coeff(a,x,n),n=0..50); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^k x^(2 k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]
%Y Cf. A000009, A052812, A255528, A305628.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Sep 10 2018
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