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%I #15 Feb 20 2022 06:43:25
%S 1,1,-4,-3,132,-375,-8298,86121,636696,-20318607,15154290,5555366289,
%T -57903946092,-1608939709767,44662076643870,329040381072825,
%U -31446740971136592,195779189199531105,21694625692807192938,-496937940680594097279
%N Expansion of e.g.f. 1/(1 - x*exp(-3*x)).
%F a(n) = n! * Sum_{k=0..n} (-3 * (n-k))^k/k!.
%F a(0) = 1 and a(n) = n * Sum_{k=0..n-1} (-3)^(n-1-k) * binomial(n-1,k) * a(k) for n > 0.
%t a[0] = 1; a[n_] := n!*Sum[(-3*(n - k))^k/k!, {k, 0, n}]; Array[a, 20, 0] (* _Amiram Eldar_, Feb 19 2022 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-3*x))))
%o (PARI) a(n) = n!*sum(k=0, n, (-3*(n-k))^k/k!);
%o (PARI) a(n) = if(n==0, 1, n*sum(k=0, n-1, (-3)^(n-1-k)*binomial(n-1, k)*a(k)));
%Y Column k=3 of A351791.
%Y Cf. A336951, A351778.
%K sign
%O 0,3
%A _Seiichi Manyama_, Feb 19 2022
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