A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Author

Ilya Gutkovskiy, May 23 2020

Keywords

• sign

Comments

Inverse binomial transform of A000240.

Crossrefs

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Programs

Mathematica

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

PARI

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

Formula

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).

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

a(n) = A000023(n) - A122803(n).

a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022

a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023