OFFSET
0,2
FORMULA
a(n) = (n!)^2 * [x^n] 1 / (1 - 2 * Sum_{k>=1} x^k / (k!)^2).
a(n) = (n!)^2 * [x^n] 1 / (3 - 2 * BesselI(0,2*sqrt(x))).
a(n) ~ (n!)^2 / (2 * BesselI(1, 2*sqrt(r)) * r^(n + 1/2)), where r = 0.4473998881770456142157108538567782213913712561... is the root of the equation 2*BesselI(0, 2*sqrt(r)) = 3. - Vaclav Kotesovec, Jul 17 2020
MATHEMATICA
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; CoefficientList[Series[1/(1 - 2 Sum[x^k/(k!)^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 12 2020
STATUS
approved