A335980 Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).

1, 3, 7, 11, 7, -5, 23, 75, -281, -101, 4663, -14229, -41721, 532667, -1464489, -8840053, 103689511, -313202725, -2348557705, 32041266859, -127039882425, -762423051013, 14393151011735, -81523161874741, -236027974047897, 8564406463119387

Offset: 0

Author

Ilya Gutkovskiy, Jul 03 2020

Keywords

• sign

Crossrefs

Cf. A001861, A035009, A109747, A213170, A335868, A335981, A335982.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[2 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

PARI

my(N=33, x='x+O('x^N)); Vec(serlaplace(exp(2 * (1 - exp(-x)) + x))) \\ Joerg Arndt, Jul 04 2020

Formula

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

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