A336635
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^2 - 1).
Original entry on oeis.org
1, 2, 14, 176, 3470, 96792, 3590048, 169686792, 9903471502, 696692504552, 57958925154584, 5614276497440712, 625153195794408608, 79159558899671117896, 11293672011942106846808, 1801015209162807119535216, 318805481931592799427378062
Offset: 0
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nmax = 16; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^2 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
A323664
Expansion of e.g.f. exp(exp(x)*BesselI(0,2*x) - 1).
Original entry on oeis.org
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
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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
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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}]
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my(x='x + O('x^25)); Vec(serlaplace(exp(exp(x)*besseli(0,2*x) - 1))) \\ Michel Marcus, Jan 24 2019
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