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A360732 Expansion of Sum_{k>0} (k * x * (1 + (k * x)^k))^k.

%I A360732 #16 Aug 09 2023 00:51:19
%S A360732 1,5,27,288,3125,48907,823543,17039360,387479538,10048828125,
%T A360732 285311670611,8929262337009,302875106592253,11116754387067959,
%U A360732 437894195556640625,18448995890703106048,827240261886336764177,39347760450413560593753
%N A360732 Expansion of Sum_{k>0} (k * x * (1 + (k * x)^k))^k.
%H A360732 Winston de Greef, <a href="/A360732/b360732.txt">Table of n, a(n) for n = 1..385</a>
%F A360732 a(n) = Sum_{d|n} d^n * binomial(d,n/d-1).
%F A360732 If p is an odd prime, a(p) = p^p.
%t A360732 a[n_] := DivisorSum[n, #^n * Binomial[#, n/# - 1] &]; Array[a, 20] (* _Amiram Eldar_, Aug 09 2023 *)
%o A360732 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+(k*x)^k))^k))
%o A360732 (PARI) a(n) = sumdiv(n, d, d^n*binomial(d, n/d-1));
%Y A360732 Cf. A360712, A360726.
%K A360732 nonn
%O A360732 1,2
%A A360732 _Seiichi Manyama_, Feb 18 2023