A352467 - OEIS

%I #6 Mar 18 2022 00:12:42

%S 1,1,37,8551,6886069,14323022551,64085654997739,545107167737695109,

%T 8062740187879748199029,193866963305030079530064391,

%U 7188682292472952994057436691387,394013888612808806428687953794890229,30829606055995735731623164115609901072859

%N a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^2 * a(n-k).

%F Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^2).

%F Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = 1 / (2 - (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2).

%t 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}]

%t 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}]

%Y Cf. A094088, A102221, A352468, A352470.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 17 2022