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a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.
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%I #14 Jan 04 2024 08:57:36

%S 0,1,6,9,28,-15,306,-1799,14904,-133407,1335070,-14684439,176214996,

%T -2290792751,32071101258,-481066515495,7697064252016,-130850092279359,

%U 2355301661034294,-44750731559644727,895014631192902540,-18795307255050944079

%N a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / 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^3.

%F E.g.f.: B_3(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, 3, stirling(3, k, 2)*x^k)*exp(x)/(1+x))))

%Y Column k=3 of A368724.

%Y Cf. A048993, A337001, A368585.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jan 04 2024