OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (phi^n * hypergeom([1/2, -n], [2], -4/phi) - (-phi)^(-n) * hypergeom([1/2, -n], [2], 4*phi))/sqrt(5), where phi = (1+sqrt(5))/2 = A001622.
Recurrence: 19*(n+1)*(n+2)*(11*n+13)*a(n) + 2*(55*n^3+208*n^2+311*n+230)*a(n+1) + 2*(55*n^3+373*n^2+674*n+206)*a(n+3) = (n+2)*(297*n^2+1022*n+617)*a(n+2) + (n+3)*(n+5)*(11*n+2)*a(n+4).
E.g.f.: 2*exp(5*x/2)*sinh(x*sqrt(5)/2)*(BesselI_0(2*x) - BesselI_1(2*x))/sqrt(5) (the product of e.g.f. for Fibonacci and Catalan numbers).
a(n) ~ (phi + 4)^(n + 3/2) / (8 * sqrt(5*Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2018
MATHEMATICA
Table[Sum[Binomial[n, k] Fibonacci[k] CatalanNumber[n - k], {k, 0, n}], {n, 0, 30}] (* or *)
Round@Table[(GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] - (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio])/Sqrt[5], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)
PROG
(PARI) for(n=0, 30, print1(sum(k=0, n, binomial(n, k)*fibonacci(k)* binomial(2*n-2*k, n-k)/(n-k+1)), ", ")) \\ G. C. Greubel, Oct 22 2018
(Magma) [(&+[Binomial(n, k)*Fibonacci(k)*Catalan(n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Oct 22 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 06 2016
STATUS
approved