A323666 - OEIS

%I #13 Jan 29 2019 05:02:22

%S 1,2,10,64,498,4544,47272,549448,7032338,98034816,1475781592,

%T 23821854808,409932257560,7483462406840,144320890075608,

%U 2929683071286416,62402858556637970,1390821290318306688,32355429437927804952,783919832399050511928,19741529222451177258920,515813862624032150918280

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

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

%p 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

%t nmax = 21; CoefficientList[Series[Exp[Exp[2 x] BesselI[0, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*besseli(0, 2*x)-1))) \\ _Michel Marcus_, Jan 24 2019

%Y Cf. A000984, A304788.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jan 23 2019