cp's OEIS Frontend

A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.

%I A368717 #20 Jul 18 2025 11:32:59
%S A368717 0,1,14,39,100,125,546,-1421,15464,-132615,1336150,-14683009,
%T A368717 176216844,-2290790411,32071104170,-481066511925,7697064256336,
%U A368717 -130850092274191,2355301661040414,-44750731559637545,895014631192910900,-18795307255050934419
%N A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.
%H A368717 Robert Israel, <a href="/A368717/b368717.txt">Table of n, a(n) for n = 0..448</a>
%H A368717 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368717 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A368717 a(0) = 0; a(n) = -n*a(n-1) + n^4.
%F A368717 E.g.f.: B_4(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
%F A368717 a(n) ~ (-1)^n * exp(-1) * n!. - _Vaclav Kotesovec_, Jul 18 2025
%p A368717 f:= proc(n) option remember;
%p A368717   - n*procname(n-1)+n^4
%p A368717 end proc:
%p A368717 f(0):= 0:
%p A368717 seq(f(i),i=0..30); # _Robert Israel_, May 13 2025
%t A368717 Table[-n + 2*n^2 + n^3 + (-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 18 2025 *)
%o A368717 (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 4, stirling(4, k, 2)*x^k)*exp(x)/(1+x))))
%Y A368717 Column k=4 of A368724.
%Y A368717 Cf. A048993, A337002, A368586.
%K A368717 sign
%O A368717 0,3
%A A368717 _Seiichi Manyama_, Jan 04 2024