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A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

%I A324158 #34 Sep 03 2019 11:21:46
%S A324158 1,2,2,6,2,23,2,50,56,107,2,660,2,499,1592,2370,2,8246,2,18557,21786,
%T A324158 11387,2,175198,43752,53419,298892,487762,2,1891098,2,2552066,3905222,
%U A324158 1114403,3785462,29081597,2,4981099,48376512,95510772,2,218764940,2,346411232,770590352
%N A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.
%H A324158 Seiichi Manyama, <a href="/A324158/b324158.txt">Table of n, a(n) for n = 1..5000</a>
%F A324158 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d+d-2,d-1).
%F A324158 a(p) = 2, where p is prime.
%t A324158 nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A324158 Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]
%o A324158 (PARI) a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ _Michel Marcus_, Sep 02 2019
%o A324158 (PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, x^k/(1-k*x^k)^k)) \\ _Seiichi Manyama_, Sep 03 2019
%Y A324158 Cf. A055225, A087909, A157019, A157020, A167531, A324159.
%K A324158 nonn
%O A324158 1,2
%A A324158 _Ilya Gutkovskiy_, Sep 02 2019