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a(n) = Sum_{k=0..n} (2*k)^(n-k).
5

%I #19 Dec 08 2021 06:49:26

%S 1,1,3,9,31,125,579,3009,17255,108005,731883,5331625,41501135,

%T 343405709,3007557523,27775308049,269603741111,2742598070709,

%U 29164361115067,323444222468089,3733412864370975,44767318872513885,556707323098632547,7168524182698345313

%N a(n) = Sum_{k=0..n} (2*k)^(n-k).

%H Seiichi Manyama, <a href="/A349970/b349970.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: Sum_{k>=0} x^k/(1 - 2*k * x).

%F a(n) ~ sqrt(Pi) * (2*n/LambertW(2*exp(1)*n))^(1/2 + n - n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - _Vaclav Kotesovec_, Dec 07 2021

%t a[n_] := Sum[If[k == n == 0, 1, (2*k)^(n - k)], {k, 0, n}]; Array[a, 24, 0] (* _Amiram Eldar_, Dec 07 2021 *)

%o (PARI) a(n) = sum(k=0, n, (2*k)^(n-k));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-2*k*x)))

%Y Cf. A349963, A349969.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 07 2021