A334190 a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).

1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2020

Keywords

• sign

Crossrefs

Column k=2 of A334192.

Cf. A007405, A009235, A293037, A308536, A334191.

Programs

Mathematica

nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)

Formula

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(2*x)) / 2).