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%I #13 Feb 07 2022 21:44:39
%S 1,3,12,48,201,885,4116,20298,106365,592455,3503532,21946620,
%T 145210305,1011726417,7400390052,56668826118,453116188821,
%U 3774297532467,32682069679548,293632972911048,2732593851548985,26299137526992525,261387306941467188,2679392140776188706
%N a(n) = Sum_{k=0..n} 3^k * k^(n-k).
%H Seiichi Manyama, <a href="/A351282/b351282.txt">Table of n, a(n) for n = 0..572</a>
%F a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/3))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/3) - n) / LambertW(exp(1)*n/3)^(n + 1/2).
%F G.f.: Sum_{k>=0} 3^k * x^k / (1 - k*x). - _Ilya Gutkovskiy_, Feb 06 2022
%t Join[{1}, Table[Sum[3^k*k^(n-k), {k, 0, n}], {n, 1, 25}]]
%o (PARI) a(n) = sum(k=0, n, 3^k*k^(n-k)); \\ _Michel Marcus_, Feb 06 2022
%Y Cf. A003101, A351279.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Feb 06 2022
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