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%I #18 Feb 07 2022 21:44:31
%S 1,2,6,18,58,202,762,3114,13754,65386,332922,1806506,10398266,
%T 63226858,404640250,2716838186,19083233210,139874994282,1067462826874,
%U 8464760754602,69620304280890,592925117961450,5220996124450042,47467755352580650,445027186867923642
%N a(n) = Sum_{k=0..n} 2^k * k^(n-k).
%H Seiichi Manyama, <a href="/A351279/b351279.txt">Table of n, a(n) for n = 0..582</a>
%F G.f.: Sum_{k>=0} (2*x)^k/(1 - k*x).
%F a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/2))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/2) - n) / LambertW(exp(1)*n/2)^(n + 1/2). - _Vaclav Kotesovec_, Feb 06 2022
%t a[0] = 1; a[n_] := Sum[2^k * k^(n-k), {k, 1, n}]; Array[a, 25, 0] (* _Amiram Eldar_, Feb 06 2022 *)
%o (PARI) a(n) = sum(k=0, n, 2^k*k^(n-k));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)))
%Y Cf. A003101, A026898, A038125, A349962.
%K nonn,easy
%O 0,2
%A _Seiichi Manyama_, Feb 06 2022