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%I #24 Jan 29 2019 05:24:25
%S 1,1,4,17,93,592,4333,35513,321422,3175143,33932527,389459534,
%T 4771856455,62099773309,854664145650,12393250075843,188732680806329,
%U 3009802364637792,50136592926632925,870386602634809233,15715357418255989580,294571161201947141223,5722457506215132179933
%N Expansion of e.g.f. exp(exp(x)*BesselI(0,2*x) - 1).
%F a(0) = 1; a(n) = Sum_{k=1..n} A002426(k)*binomial(n-1,k-1)*a(n-k).
%p 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
%t nmax = 22; CoefficientList[Series[Exp[Exp[x] BesselI[0, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t 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}]
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(exp(x)*besseli(0,2*x) - 1))) \\ _Michel Marcus_, Jan 24 2019
%Y Cf. A002426, A323666.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 23 2019
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