A352467 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^2 * a(n-k).
1, 1, 37, 8551, 6886069, 14323022551, 64085654997739, 545107167737695109, 8062740187879748199029, 193866963305030079530064391, 7188682292472952994057436691387, 394013888612808806428687953794890229, 30829606055995735731623164115609901072859
Offset: 0
Keywords
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 12}] nmax = 24; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]
Formula
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^2).
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (2 - (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2).