A308846 Expansion of e.g.f. x*exp(-x) / BesselI(1,2*x).

1, -1, 0, 2, -1, -11, 11, 125, -181, -2443, 4534, 73116, -164075, -3106169, 8150624, 177689590, -533231545, -13167063415, 44461467464, 1226832808294, -4603245727023, -140381127928757, 579412786145335, 19352381237997941, -87137414891475575, -3163451350469601721, 15431005593881754746

Offset: 0

Author

Ilya Gutkovskiy, Jun 28 2019

Keywords

• sign

Comments

E.g.f. is inverse of e.g.f. for A001006 (Motzkin numbers).

Crossrefs

Cf. A001006, A292440.

Programs

Mathematica

nmax = 26; CoefficientList[Series[x Exp[-x]/BesselI[1, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 26}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(-x) / besseli(1,2*x))) \\ Michel Marcus, Jul 02 2019

Formula

E.g.f.: 1 / Sum_{k>=0} A001006(k)*x^k/k!.