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%I #6 Feb 15 2021 17:59:14
%S 1,0,-4,-5,22,98,-5,-1458,-5136,9053,161328,549822,-1954067,-30099188,
%T -114161728,500200027,8875931202,42311243830,-149028931789,
%U -3816065804086,-24704581255020,33033659868037,2184285021783940,20047242475274290,30117550563701293
%N E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.
%F a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * Stirling2(k, 2).
%F a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * (2^(k-1) - 1).
%F a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A000587(k) * A000587(n-k).
%t nmax = 26; CoefficientList[Series[(Exp[1 - Exp[x]] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
%t Table[Sum[(-1)^k StirlingS2[n, k] StirlingS2[k, 2], {k, 2, n}], {n, 2, 26}]
%Y Cf. A000225, A000558, A000587, A008277, A213170, A309775.
%K sign
%O 2,3
%A _Ilya Gutkovskiy_, Feb 15 2021
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