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Expansion of Sum_{k>0} (k * x * (1 + (2 * x)^k))^k.
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%I #13 Aug 02 2023 02:00:31

%S 1,6,27,288,3125,47368,823543,16793600,387425673,10000500000,

%T 285311670611,8916118771200,302875106592253,11112007563452544,

%U 437893890412859375,18446744108073484288,827240261886336764177,39346408077084637733376

%N Expansion of Sum_{k>0} (k * x * (1 + (2 * x)^k))^k.

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

%F If p is an odd prime, a(p) = p^p.

%t a[n_] := DivisorSum[n, 2^(n-#) * #^# * Binomial[#, n/# - 1] &]; Array[a, 20] (* _Amiram Eldar_, Aug 02 2023 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+(2*x)^k))^k))

%o (PARI) a(n) = sumdiv(n, d, 2^(n-d)*d^d*binomial(d, n/d-1));

%Y Cf. A360732.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 19 2023