A277251 Exponential convolution of Lucas (A000032) and Catalan (A000108) numbers.

2, 3, 9, 29, 107, 430, 1840, 8230, 38015, 179873, 867079, 4242111, 21006358, 105072063, 530058079, 2693632580, 13775807415, 70847283680, 366167521240, 1900884870494, 9907318315587, 51822028122623, 271949090063769, 1431369293422604, 7554372307564282

Offset: 0

Author

Vladimir Reshetnikov, Oct 06 2016

Keywords

• nonn

Crossrefs

Cf. A277220, A000032, A000108, A014330, A014334, A090826.

Programs

Mathematica

Table[Sum[Binomial[n, k] LucasL[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], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000032(k) * A000108(n-k).

a(n) = phi^n * hypergeom([1/2, -n], [2], -4/phi) + (-phi)^(-n) * hypergeom([1/2, -n], [2], 4*phi), 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)*cosh(x*sqrt(5)/2)*(BesselI_0(2*x) - BesselI_1(2*x)) (the product of e.g.f. for Lucas and Catalan numbers).

a(n) ~ (phi + 4)^(n + 3/2) / (8 * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2018