A341586 E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.

1, 0, -4, -5, 22, 98, -5, -1458, -5136, 9053, 161328, 549822, -1954067, -30099188, -114161728, 500200027, 8875931202, 42311243830, -149028931789, -3816065804086, -24704581255020, 33033659868037, 2184285021783940, 20047242475274290, 30117550563701293

Offset: 2

Author

Ilya Gutkovskiy, Feb 15 2021

Keywords

• sign

Crossrefs

Cf. A000225, A000558, A000587, A008277, A213170, A309775.

Programs

Mathematica

nmax = 26; CoefficientList[Series[(Exp[1 - Exp[x]] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[(-1)^k StirlingS2[n, k] StirlingS2[k, 2], {k, 2, n}], {n, 2, 26}]

Formula

a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * Stirling2(k, 2).

a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * (2^(k-1) - 1).

a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A000587(k) * A000587(n-k).