A346220 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2 ).

1, 1, 2, 7, 40, 321, 3356, 45123, 752256, 15018433, 355378732, 9823042923, 311510611072, 11242338245009, 458052976883672, 20851748359005567, 1054108827258438656, 58860837547461314049, 3606677286494115444812, 241397002229305033296603, 17579096976247770110062080

Offset: 0

Author

Ilya Gutkovskiy, Jul 11 2021

Keywords

• nonn

Crossrefs

Cf. A003724, A023998.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[x]] - BesselJ[0, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, 2 k + 1]^2 (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 20}]

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / ((2*n + 1)!)^2 ).

a(0) = 1; a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * (2*k+1) * a(n-2*k-1).