A368717 - OEIS

%I #13 Jan 04 2024 08:57:31

%S 0,1,14,39,100,125,546,-1421,15464,-132615,1336150,-14683009,

%T 176216844,-2290790411,32071104170,-481066511925,7697064256336,

%U -130850092274191,2355301661040414,-44750731559637545,895014631192910900,-18795307255050934419

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>

%F a(0) = 0; a(n) = -n*a(n-1) + n^4.

%F E.g.f.: B_4(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

%o (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 Column k=4 of A368724.

%Y Cf. A048993, A337002, A368586.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jan 04 2024