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a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
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%I #8 Jan 27 2021 07:29:09

%S 1,-1,-2,9,32,-285,-1236,18725,86176,-2087001,-9204580,351964569,

%T 1336442304,-83422970917,-231889447076,26389118293005,35917342192064,

%U -10722110983670193,5028963509133756,5432569724760331841,-14852185163192897120,-3352369390318855889661

%N a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

%F E.g.f.: 1 / (1 + exp(x) * x * (1 + x)).

%F E.g.f.: 1 / (1 + Sum_{k>=1} k^2 * x^k / k!).

%t a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

%t nmax = 21; CoefficientList[Series[1/(1 + Exp[x] x (1 + x)), {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A302189, A302397, A308861, A316087, A335578.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jan 26 2021