login
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^3 - 1).
2

%I #5 Jul 28 2020 22:20:41

%S 1,3,33,660,20817,935388,56149098,4311694467,410200118577,

%T 47174279349540,6431874002292978,1023398757621960327,

%U 187566773426941146498,39164789611542644630415,9229712819952662426436507,2435069724188535096598261305

%N Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^3 - 1).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * A002893(k) * k * a(n-k).

%t nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^3 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2

%t a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

%Y Cf. A002893, A023998, A247452, A336635, A336637.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 28 2020