OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - Sum_{n>=1} x^n / ((n-1)!)^2).
a(0) = 1; a(n) = Sum_{k=0..n-1} (binomial(n,k) * (n-k))^2 * a(k).
a(n) ~ n!^2 / ((1 + r^(3/2)*BesselI(1, 2*sqrt(r))) * r^n), where r = 0.592860029867912878114616561736048937618032595935338954527835... is the root of the equation r*BesselI(0, 2*sqrt(r)) = 1. - Vaclav Kotesovec, May 04 2024
MATHEMATICA
nmax = 16; CoefficientList[Series[1/(1 - x BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(Binomial[n, k] (n - k))^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
PROG
(PARI) a(n) = {n!^2*polcoef(1 / (1 - sum(k=1, n, x^k / ((k-1)!)^2) + O(x*x^n)), n)} \\ Andrew Howroyd, Mar 04 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 04 2021
STATUS
approved