A323664 Expansion of e.g.f. exp(exp(x)*BesselI(0,2*x) - 1).

1, 1, 4, 17, 93, 592, 4333, 35513, 321422, 3175143, 33932527, 389459534, 4771856455, 62099773309, 854664145650, 12393250075843, 188732680806329, 3009802364637792, 50136592926632925, 870386602634809233, 15715357418255989580, 294571161201947141223, 5722457506215132179933

Offset: 0

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

• nonn

Crossrefs

Cf. A002426, A323666.

Programs

Maple

seq(n!*coeff(series(exp(exp(x)*BesselI(0,2*x)-1),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019

Mathematica

nmax = 22; CoefficientList[Series[Exp[Exp[x] BesselI[0, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[3^k Hypergeometric2F1[1/2, -k, 1, 4/3] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

PARI

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A002426(k)*binomial(n-1,k-1)*a(n-k).