A330497 - OEIS

%I #19 Sep 08 2022 08:46:24

%S 1,0,1,26,1089,70124,6495985,821315214,136115947009,28651724077976,

%T 7470040450004001,2363470644596843330,892244303052345224641,

%U 396227360441775922668036,204487588996059177697597969,121370399839482643287189048374

%N a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

%F a(n) = n! * [x^n] exp(-x/(1 - n*x)) / (1 - n*x).

%F a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k)^2 * n^k * k!.

%F a(n) ~ sqrt(2*Pi) * BesselJ(0,2) * n^(2*n + 1/2) / exp(n). - _Vaclav Kotesovec_, Dec 18 2019

%t Join[{1}, Table[n! Sum[(-1)^k 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 (Magma) [Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // _Marius A. Burtea_, Dec 18 2019

%Y Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Dec 18 2019