OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
FORMULA
a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).
a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.
a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).
a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).
a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017
MATHEMATICA
Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 05 2017
STATUS
approved