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%I #13 Jan 29 2019 05:02:08
%S 1,3,19,152,1467,16445,208471,2934321,45254447,756995131,13623709401,
%T 262067291106,5358900661509,115953603121881,2644399031839729,
%U 63346390393538780,1589177904965680263,41642328796769014811,1137083068108603968349,32287430515011314674632,951565685429585731747913
%N Expansion of e.g.f. exp(exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - 1).
%F a(0) = 1; a(n) = Sum_{k=1..n} A001700(k)*binomial(n-1,k-1)*a(n-k).
%p seq(n!*coeff(series(exp(exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))-1),x=0,21),x,n),n=0..20); # _Paolo P. Lava_, Jan 28 2019
%t nmax = 20; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] + BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Binomial[2 k + 1, k + 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*(besseli(0, 2*x)+x*besseli(1, 2*x))-1))) \\ _Michel Marcus_, Jan 24 2019
%Y Cf. A001700, A302197, A304788.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 23 2019