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%I #13 Jan 29 2019 05:02:14
%S 1,1,3,10,43,211,1191,7463,51535,386809,3133273,27184620,251253157,
%T 2461527511,25459020289,276987375642,3160197122183,37705878268985,
%U 469340324930493,6081394853597162,81866045488063721,1142928276326927223,16521454311961005245,246917508673451732077
%N Expansion of e.g.f. exp(BesselI(0,2*x) + BesselI(1,2*x) - 1).
%F a(0) = 1; a(n) = Sum_{k=1..n} A001405(k)*binomial(n-1,k-1)*a(n-k).
%p seq(n!*coeff(series(exp(BesselI(0,2*x)+BesselI(1,2*x)-1),x=0,24),x,n),n=0..23); # _Paolo P. Lava_, Jan 28 2019
%t nmax = 23; CoefficientList[Series[Exp[BesselI[0, 2 x] + BesselI[1, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(besseli(0, 2*x)+x*besseli(1, 2*x)-1))) \\ _Michel Marcus_, Jan 24 2019
%Y Cf. A001405, A302197, A304788.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 23 2019
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