A335501 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

1, 2, 10, 86, 1098, 19142, 431926, 12150518, 414474570, 16781350694, 792845706630, 43107783435158, 2666346336398454, 185796230244565462, 14464057604306584774, 1248919312238777955086, 118855834572748011228490, 12397162719421869533115622

Offset: 0

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

• nonn

Crossrefs

Cf. A001861, A023998.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = (2/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[2 Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] exp(2 * Sum_{k>=1} x^k / (k!)^2).

a(n) = (n!)^2 * [x^n] exp(2 * (BesselI(0,2*sqrt(x)) - 1)).