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A360726 Expansion of Sum_{k>0} (k * x * (1 + x^k))^k.

%I A360726 #19 Aug 02 2023 02:00:24
%S A360726 1,5,27,264,3125,46741,823543,16778240,387420570,10000015625,
%T A360726 285311670611,8916100729755,302875106592253,11112006831322817,
%U A360726 437893890380890625,18446744073843770368,827240261886336764177,39346408075300025059665
%N A360726 Expansion of Sum_{k>0} (k * x * (1 + x^k))^k.
%H A360726 Winston de Greef, <a href="/A360726/b360726.txt">Table of n, a(n) for n = 1..385</a>
%F A360726 a(n) = Sum_{d|n} d^d * binomial(d,n/d-1).
%F A360726 If p is an odd prime, a(p) = p^p.
%t A360726 a[n_] := DivisorSum[n, #^# * Binomial[#, n/# - 1] &]; Array[a, 20] (* _Amiram Eldar_, Aug 02 2023 *)
%o A360726 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+x^k))^k))
%o A360726 (PARI) a(n) = sumdiv(n, d, d^d*binomial(d, n/d-1));
%Y A360726 Cf. A360611, A360727, A360728.
%Y A360726 Cf. A360712, A360732.
%K A360726 nonn
%O A360726 1,2
%A A360726 _Seiichi Manyama_, Feb 18 2023