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%I #9 May 08 2024 08:52:37
%S 0,0,1,0,-5,-10,16,154,365,-750,-9749,-35222,20956,1013220,6007821,
%T 10272092,-129948837,-1405396426,-6318145964,7407235766,371429230721,
%U 3172609248526,11070816858267,-73488239926510,-1500342260080360,-11917913896465720,-31231507292803479
%N Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.
%F a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * binomial(k,2).
%F a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,2) * A000587(n-k).
%t nmax = 26; CoefficientList[Series[Exp[1 - Exp[x]] (Exp[x] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^k StirlingS2[n, k] Binomial[k, 2], {k, 0, n}], {n, 0, 26}]
%Y Cf. A000217, A000225, A000587, A003128, A059606, A081047, A101851.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, May 07 2024