A323666 Expansion of e.g.f. exp(exp(2*x)*BesselI(0,2*x) - 1).

1, 2, 10, 64, 498, 4544, 47272, 549448, 7032338, 98034816, 1475781592, 23821854808, 409932257560, 7483462406840, 144320890075608, 2929683071286416, 62402858556637970, 1390821290318306688, 32355429437927804952, 783919832399050511928, 19741529222451177258920, 515813862624032150918280

Offset: 0

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

• nonn

Crossrefs

Cf. A000984, A304788.

Programs

Maple

seq(n!*coeff(series(exp(exp(2*x)*BesselI(0,2*x)-1),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 28 2019

Mathematica

nmax = 21; CoefficientList[Series[Exp[Exp[2 x] BesselI[0, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[2 k, k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

PARI

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000984(k)*binomial(n-1,k-1)*a(n-k).