A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

0, 0, 1, 0, -5, -10, 16, 154, 365, -750, -9749, -35222, 20956, 1013220, 6007821, 10272092, -129948837, -1405396426, -6318145964, 7407235766, 371429230721, 3172609248526, 11070816858267, -73488239926510, -1500342260080360, -11917913896465720, -31231507292803479

Offset: 0

Author

Ilya Gutkovskiy, May 07 2024

Keywords

• sign

Crossrefs

Cf. A000217, A000225, A000587, A003128, A059606, A081047, A101851.

Programs

Mathematica

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * binomial(k,2).

a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,2) * A000587(n-k).