A328004 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(2*k,k) * x^k / k!).
1, 2, 14, 140, 1854, 30692, 609812, 14135816, 374486782, 11161030388, 369597971484, 13463177200376, 535000400076660, 23031528320070584, 1067766010124118200, 53038672987708575920, 2810204538580052967422, 158202066016882053997204, 9429962256806049820343564
Offset: 0
Keywords
Programs
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Mathematica
nmax = 18; CoefficientList[Series[1/(2 - Exp[2 x] BesselI[0, 2 x]), {x, 0, nmax}], x] Range[0, nmax]! 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}] -
PARI
my(x='x+O('x^25)); Vec(serlaplace(1/(2 - exp(2*x) * (besseli(0,2*x))))) \\ Michel Marcus, Oct 02 2019
Formula
E.g.f.: 1 / (2 - exp(2*x) * BesselI(0,2*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000984(k) * a(n-k).
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
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