A318183 - OEIS

%I #15 Feb 16 2025 08:33:56

%S 1,1,-1,1,25,-674,15211,-331827,5987745,15901597,-13125035449,

%T 1292056076070,-103145930581319,7462324963409941,-464957409070517453,

%U 16313974895147212801,2059903411953959582849,-708700955022151333496910,143215213612865558214820303,-24681846509158429152517973103

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

%H Seiichi Manyama, <a href="/A318183/b318183.txt">Table of n, a(n) for n = 0..282</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>

%F a(n) = n! * [x^n] exp((1 - exp(-n*x))/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_, Aug 20 2018

%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[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]

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

%t Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* _Peter Luschny_, Aug 20 2018 *)

%o (PARI) {a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jul 27 2019

%Y Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

%K sign,changed

%O 0,5

%A _Ilya Gutkovskiy_, Aug 20 2018