A352465 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2*n,2*k)^2 * k * a(n-k).
1, 1, 19, 1576, 356035, 172499176, 154989443170, 234120771123513, 553941959716031715, 1945912976888526218512, 9731900583801946493234794, 66990924607889809703423378253, 617312916540194845307221190273098, 7439659538258619452171059589120614701
Offset: 0
Keywords
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^2 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}] nmax = 26; Take[CoefficientList[Series[Exp[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 = exp( Sum_{n>=1} x^(2*n) / (2*n)!^2 ).
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = exp( (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2 - 1 ).