%I #14 May 05 2023 01:39:01

%S 1,3,8,35,154,756,3828,20163,108680,598026,3342404,18929092,108374252,

%T 626264700,3647936160,21396522915,126262239570,749087596620,

%U 4465444206300,26733390275130,160663411399920,968937572793060,5862111195487560,35569106862459300,216395609659221564

%N Product of Lucas and Catalan numbers: a(n) = A000032(n+1)*A000108(n).

%F G.f.: (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2))/2 )) / x.

%F 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 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 Sum_{n>=0} a(n)/8^n = 8 - 2*sqrt(10). - _Amiram Eldar_, May 05 2023

%e G.f.: A(x) = 1 + 3*x + 8*x^2 + 35*x^3 + 154*x^4 + 756*x^5 + 3828*x^6 +...

%e such that the coefficients equal the term-wise products:

%e A = [1*1, 3*1, 4*2, 7*5, 11*14, 18*42, 29*132, 47*429, 76*1430, ...].

%t a[n_] := LucasL[n+1] * CatalanNumber[n]; Array[a, 25, 0] (* _Amiram Eldar_, May 05 2023 *)

%o (PARI) {a(n)=(2*fibonacci(n)+fibonacci(n+1))*binomial(2*n,n)/(n+1)}

%o (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 for(n=0,25,print1(a(n),", "))

%Y Cf. A098614, A215931, A098616, A000032, A000108.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 08 2012