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%I #13 Jan 13 2024 06:48:59
%S 0,1,34,345,2404,15145,98646,707329,5691400,51281649,512916490,
%T 5642242441,67707158124,880193426905,12322708514494,184840628476785,
%U 2957450056677136,50276650964931169,904979717370650610,17194614630044837689,343892292600899953780
%N a(n) = n! * Sum_{k=0..n} k^5 / k!.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F a(0) = 0; a(n) = n*a(n-1) + n^5.
%F E.g.f.: B_5(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials.
%F a(n) ~ 52*exp(1)*n!. - _Vaclav Kotesovec_, Jan 13 2024
%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 5, stirling(5, k, 2)*x^k)*exp(x)/(1-x))))
%Y Column k=5 of A337085.
%Y Cf. A048993, A368718.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jan 04 2024
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