A323668 Expansion of e.g.f. exp(exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - 1).

1, 3, 19, 152, 1467, 16445, 208471, 2934321, 45254447, 756995131, 13623709401, 262067291106, 5358900661509, 115953603121881, 2644399031839729, 63346390393538780, 1589177904965680263, 41642328796769014811, 1137083068108603968349, 32287430515011314674632, 951565685429585731747913

Offset: 0

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

• nonn

Crossrefs

Cf. A001700, A302197, A304788.

Programs

Maple

seq(n!*coeff(series(exp(exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))-1),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 28 2019

Mathematica

nmax = 20; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] + BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[2 k + 1, k + 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

PARI

my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*(besseli(0, 2*x)+x*besseli(1, 2*x))-1))) \\ Michel Marcus, Jan 24 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A001700(k)*binomial(n-1,k-1)*a(n-k).