A336610 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-sqrt(x) * BesselI(1,2*sqrt(x))).

1, -1, 0, 9, -4, -625, -906, 145187, 1350040, -71822385, -2093778910, 49843036199, 4422338360340, 7491520000835, -11939082153832302, -455740256735697165, 33146485198521406064, 4039886119274766333343, 2019781328116371668154

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

• sign

Crossrefs

Cf. A003725, A292952, A302397, A336209, A336227.

Programs

Mathematica

nmax = 18; CoefficientList[Series[Exp[-Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]

Formula

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