This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277251 #6 Mar 10 2018 06:44:49
%S A277251 2,3,9,29,107,430,1840,8230,38015,179873,867079,4242111,21006358,
%T A277251 105072063,530058079,2693632580,13775807415,70847283680,366167521240,
%U A277251 1900884870494,9907318315587,51822028122623,271949090063769,1431369293422604,7554372307564282
%N A277251 Exponential convolution of Lucas (A000032) and Catalan (A000108) numbers.
%F A277251 a(n) = Sum_{k=0..n} binomial(n,k) * A000032(k) * A000108(n-k).
%F A277251 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.
%F A277251 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).
%F A277251 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).
%F A277251 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
%t A277251 Table[Sum[Binomial[n, k] LucasL[k] CatalanNumber[n - k], {k, 0, n}], {n, 0,
%t A277251 30}] (* or *)
%t A277251 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 *)
%Y A277251 Cf. A277220, A000032, A000108, A014330, A014334, A090826.
%K A277251 nonn
%O A277251 0,1
%A A277251 _Vladimir Reshetnikov_, Oct 06 2016