A351279 a(n) = Sum_{k=0..n} 2^k * k^(n-k).
1, 2, 6, 18, 58, 202, 762, 3114, 13754, 65386, 332922, 1806506, 10398266, 63226858, 404640250, 2716838186, 19083233210, 139874994282, 1067462826874, 8464760754602, 69620304280890, 592925117961450, 5220996124450042, 47467755352580650, 445027186867923642
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..582
Programs
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Mathematica
a[0] = 1; a[n_] := Sum[2^k * k^(n-k), {k, 1, n}]; Array[a, 25, 0] (* Amiram Eldar, Feb 06 2022 *) -
PARI
a(n) = sum(k=0, n, 2^k*k^(n-k));
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PARI
my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)))
Formula
G.f.: Sum_{k>=0} (2*x)^k/(1 - k*x).
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