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%I #24 Dec 07 2021 10:49:52
%S 1,2,20,288,5664,141600,4298944,153638912,6319260672,294044152320,
%T 15272286131200,875880428003328,54976337351106560,3748609104907476992,
%U 275924407293425336320,21806398621389422592000,1841661678145084557099008,165530736067119754944577536
%N a(n) = Sum_{k=0..n} (2*k)^n.
%F G.f.: Sum_{k>=0} (2*k * x)^k/(1 - 2*k * x).
%F a(n) = 2^n * A031971(n).
%F a(n) ~ c * 2^n * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - _Vaclav Kotesovec_, Dec 07 2021
%t a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Dec 07 2021 *)
%o (PARI) a(n) = sum(k=0, n, (2*k)^n);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k*x)^k/(1-2*k*x)))
%Y Cf. A031971, A185393, A249459, A349970.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Dec 07 2021