|
%I #29 Sep 08 2022 08:46:24
%S 1,2,17,352,13505,830126,74717857,9263893892,1513712421377,
%T 315230799073690,81499084718806001,25612081645835777192,
%U 9615370149488574778177,4250194195208050117007942,2184834047906975645398282625,1292386053018890618812398220876
%N a(n) = n! * Sum_{k=0..n} binomial(n,k) * n^(n - k) / k!.
%H Seiichi Manyama, <a href="/A330260/b330260.txt">Table of n, a(n) for n = 0..232</a>
%F a(n) = n! * [x^n] exp(x/(1 - n*x)) / (1 - n*x).
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * k!.
%F a(n) ~ sqrt(2*Pi) * BesselI(0,2) * n^(2*n + 1/2) / exp(n). - _Vaclav Kotesovec_, Dec 18 2019
%t Join[{1}, Table[n! Sum[Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
%t Join[{1}, Table[n^n n! LaguerreL[n, -1/n], {n, 1, 15}]]
%t Table[n! SeriesCoefficient[Exp[x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]
%o (PARI) a(n) = n! * sum(k=0, n, binomial(n,k) * n^(n-k)/k!); \\ _Michel Marcus_, Dec 18 2019
%o (Magma) [Factorial(n)*&+[Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // _Marius A. Burtea_, Dec 18 2019
%Y Cf. A002720, A025167, A061711, A102757, A102773, A187021, A277373, A277452, A293146, A330497.
%Y Main diagonal of A341014.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 18 2019
|
