A301419 - OEIS

%I #17 Jun 06 2022 11:44:32

%S 1,1,3,19,201,3176,69823,2026249,74565473,3376695763,183991725451,

%T 11854772145800,890415496931689,77023751991841669,7592990698770559111,

%U 845240026276785888451,105409073489605774592897,14625467507717709778793020,2244123413703647502288608467,378751257186051653931253015229

%N a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - n*j*x).

%H Muniru A Asiru, <a href="/A301419/b301419.txt">Table of n, a(n) for n = 0..101</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = n! * [x^n] exp((exp(n*x) - 1)/n), for n > 0.

%F a(n) = Sum_{k=0..n} n^(n-k)*Stirling2(n,k).

%F a(n) = n^n * BellPolynomial(n, 1/n) for n >= 1. - _Peter Luschny_, Dec 22 2021

%F a(n) ~ exp(n/LambertW(n^2) - n) * n^(2*n) / (sqrt(1 + LambertW(n^2)) * LambertW(n^2)^n). - _Vaclav Kotesovec_, Jun 06 2022

%t Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]

%t Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]

%t Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]

%t (* Or: *)

%t A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];

%t Table[A301419[n], {n, 0, 19}] (* _Peter Luschny_, Dec 22 2021 *)

%o (GAP) List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # _Muniru A Asiru_, Mar 20 2018

%o (PARI) a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ _Michel Marcus_, Mar 23 2018

%Y Cf. A000110, A004211, A004212, A004213, A005011, A005012, A008277, A075506, A075507, A075508, A075509, A242817, A292914, A318183.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 20 2018