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Expansion of e.g.f. exp(1 - exp(-x)) * (exp(-x) - 1) * (exp(-x) - 2).
0

%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