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%I #9 Oct 02 2019 08:21:52
%S 1,2,14,140,1854,30692,609812,14135816,374486782,11161030388,
%T 369597971484,13463177200376,535000400076660,23031528320070584,
%U 1067766010124118200,53038672987708575920,2810204538580052967422,158202066016882053997204,9429962256806049820343564
%N Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(2*k,k) * x^k / k!).
%F E.g.f.: 1 / (2 - exp(2*x) * BesselI(0,2*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000984(k) * a(n-k).
%F a(n) ~ n! / ((4 + 2*exp(2*r)*BesselI(1, 2*r)) * r^(n+1)), where r = 0.30197758068953447339121214393882523964817455046976015309132... is the root of the equation exp(2*r) * BesselI(0, 2*r) = 2. - _Vaclav Kotesovec_, Oct 02 2019
%t nmax = 18; CoefficientList[Series[1/(2 - Exp[2 x] BesselI[0, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(2 - exp(2*x) * (besseli(0,2*x))))) \\ _Michel Marcus_, Oct 02 2019
%Y Cf. A000984, A075436, A308847, A328006.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 01 2019