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

0,1

Links

Table of n, a(n) for n=0..24.

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

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 *)

Crossrefs

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

Sequence in context: A213943 A153701 A176678 * A275165 A073950 A281270

Adjacent sequences: A277248 A277249 A277250 * A277252 A277253 A277254

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 06 2016

Status

approved