A277220 Exponential convolution of Fibonacci (A000045) and Catalan (A000108) numbers.

0, 1, 3, 11, 43, 180, 790, 3590, 16745, 79705, 385615, 1890747, 9375216, 46931897, 236873261, 1204089630, 6159064015, 31678706490, 163739008070, 850051218980, 4430529313065, 23175017046351, 121617754070653, 640122809255716, 3378402106118508, 17875011275340275

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000045(k) * A000108(n-k).

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

Cf. A277251, A000045, A000108, A014330, A014334, A090826.

Sequence in context: A151096 A151097 A151098 * A338999 A151099 A049184

Adjacent sequences: A277217 A277218 A277219 * A277221 A277222 A277223

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 06 2016

Status

approved