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A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).

%I A293146 #20 Feb 03 2021 08:55:26
%S A293146 1,1,5,73,2161,108101,8201701,878797165,126422091713,23514740267401,
%T A293146 5492576235204901,1574136880033408241,543143967119720304625,
%U A293146 222106209904092987888013,106221716052645457812866501,58741017143127754662557082901,37194600833984874761008613195521
%N A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).
%H A293146 Robert Israel, <a href="/A293146/b293146.txt">Table of n, a(n) for n = 0..232</a>
%F A293146 a(n) ~ BesselI(1, 2) * sqrt(2*Pi) * n^(2*n-1/2) / exp(n). - _Vaclav Kotesovec_, Oct 01 2017
%F A293146 a(n) = n! * Sum_{k=1..n} n^(n-k) * binomial(n-1,k-1)/k! for n > 0. - _Seiichi Manyama_, Feb 03 2021
%p A293146 S:=series(exp(x/(1-n*x)),x,31):
%p A293146 seq(coeff(S,x,n)*n!,n=0..30); # _Robert Israel_, Oct 01 2017
%t A293146 Table[n! SeriesCoefficient[Exp[x/(1 - n x)], {x, 0, n}], {n, 0, 16}]
%t A293146 Join[{1}, Table[n! SeriesCoefficient[Product[Exp[n^k x^(k + 1)], {k, 0, n}], {x, 0, n}], {n, 1, 16}]]
%t A293146 Join[{1}, Table[Sum[n^(n - k) n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 16}]]
%t A293146 Join[{1}, Table[n^n (n - 1)! Hypergeometric1F1[1 - n, 2, -1/n], {n, 1, 16}]]
%o A293146 (PARI) {a(n) = if(n==0, 1, n!*sum(k=1, n, n^(n-k)*binomial(n-1, k-1)/k!))} \\ _Seiichi Manyama_, Feb 03 2021
%Y A293146 Cf. A000262, A025168, A293145.
%K A293146 nonn
%O A293146 0,3
%A A293146 _Ilya Gutkovskiy_, Oct 01 2017