%I #20 May 18 2024 15:08:09
%S 0,1,3,-2,-11,31,14,-349,1047,820,-21265,90355,-26352,-2086083,
%T 14092615,-32449650,-241320287,3080629195,-15455723498,-2456654665,
%U 760213889483,-7097893818852,28459679925187,125560349169887,-3153253543188992,26852335900600041,-86130449768002245
%N Expansion of e.g.f. exp(1 - exp(-x)) * (exp(-x) - 1) * (exp(-x) - 2).
%F G.f.: Sum_{k>=0} k^2 * x^k / Product_{j=0..k} (1 + j*x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * k^2.
%t nmax = 26; CoefficientList[Series[Exp[1 - Exp[-x]] (Exp[-x] - 1) (Exp[-x] - 2), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^(n - k) StirlingS2[n, k] k^2, {k, 0, n}], {n, 0, 26}]
%Y Cf. A000587, A033452, A045406, A101851, A151881.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, May 14 2024