A336637 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^4 - 1).

1, 4, 60, 1648, 71612, 4448384, 370135632, 39480942848, 5227020747708, 837878205997216, 159457868003008640, 35459969754432262208, 9093585253916177728592, 2659611377508767798535488, 878768601771275773332660736, 325350340926291926090183214848

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

• nonn

Crossrefs

Cf. A002895, A023998, A336635, A336636.

Programs

Mathematica

nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^4 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * A002895(k) * k * a(n-k).