A308645 Expansion of e.g.f. exp(1 + x - exp(2*x)).

1, -1, -3, 3, 41, 87, -571, -5701, -14575, 156655, 2094925, 9148851, -63364423, -1474212665, -11494853995, 10945362411, 1520718442785, 20719421344991, 100137575499165, -1638818071763869, -45333849658449847, -512404024891840969, -577060092568365467, 99142586163648127771

Offset: 0

Author

Ilya Gutkovskiy, Jun 13 2019

Keywords

• sign

Crossrefs

Cf. A000587, A055882, A126390, A308536.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Exp[1 + x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Exp[1] Sum[(-1)^k (2 k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 23}]
Table[Sum[Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(2*k + 1)^n/k!.

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