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 A216541 #14 May 05 2023 01:39:01
%S A216541 1,3,8,35,154,756,3828,20163,108680,598026,3342404,18929092,108374252,
%T A216541 626264700,3647936160,21396522915,126262239570,749087596620,
%U A216541 4465444206300,26733390275130,160663411399920,968937572793060,5862111195487560,35569106862459300,216395609659221564
%N A216541 Product of Lucas and Catalan numbers: a(n) = A000032(n+1)*A000108(n).
%F A216541 G.f.: (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2))/2 )) / x.
%F A216541 G.f. satisfies: A(x) = (2+5*x) - (1+4*x)*A(x) + x*(5+2*x)*A(x)^2 - 4*x^2*A(x)^3 + x^3*A(x)^4.
%F A216541 n*(n+1)*a(n) -2*n*(2n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Sep 11 2012
%F A216541 Sum_{n>=0} a(n)/8^n = 8 - 2*sqrt(10). - _Amiram Eldar_, May 05 2023
%e A216541 G.f.: A(x) = 1 + 3*x + 8*x^2 + 35*x^3 + 154*x^4 + 756*x^5 + 3828*x^6 +...
%e A216541 such that the coefficients equal the term-wise products:
%e A216541 A = [1*1, 3*1, 4*2, 7*5, 11*14, 18*42, 29*132, 47*429, 76*1430, ...].
%t A216541 a[n_] := LucasL[n+1] * CatalanNumber[n]; Array[a, 25, 0] (* _Amiram Eldar_, May 05 2023 *)
%o A216541 (PARI) {a(n)=(2*fibonacci(n)+fibonacci(n+1))*binomial(2*n,n)/(n+1)}
%o A216541 (PARI) {a(n)=polcoeff( (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2 +x^2*O(x^n)))/2 )) / x,n)}
%o A216541 for(n=0,25,print1(a(n),", "))
%Y A216541 Cf. A098614, A215931, A098616, A000032, A000108.
%K A216541 nonn
%O A216541 0,2
%A A216541 _Paul D. Hanna_, Sep 08 2012