A324158 - OEIS

%I #34 Sep 03 2019 11:21:46

%S 1,2,2,6,2,23,2,50,56,107,2,660,2,499,1592,2370,2,8246,2,18557,21786,

%T 11387,2,175198,43752,53419,298892,487762,2,1891098,2,2552066,3905222,

%U 1114403,3785462,29081597,2,4981099,48376512,95510772,2,218764940,2,346411232,770590352

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

%H Seiichi Manyama, <a href="/A324158/b324158.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d+d-2,d-1).

%F a(p) = 2, where p is prime.

%t nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]

%o (PARI) a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ _Michel Marcus_, Sep 02 2019

%o (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 Cf. A055225, A087909, A157019, A157020, A167531, A324159.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Sep 02 2019