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 A068767 #21 Feb 03 2025 08:42:16
%S A068767 1,1,10,105,1150,13050,152500,1825625,22293750,276758750,3483287500,
%T A068767 44352006250,570333187500,7396680812500,96638930625000,
%U A068767 1270796364765625,16806545339843750,223400240246093750
%N A068767 Generalized Catalan numbers 5*x*A(x)^2 -A(x) +1 -4*x=0.
%C A068767 a(n) = K(5,5; n)/5 with K(a,b; n) defined in a comment to A068763.
%H A068767 Fung Lam, <a href="/A068767/b068767.txt">Table of n, a(n) for n = 0..860</a>
%F A068767 a(n) = (5^n) * p(n, -4/5) with the row polynomials p(n, x) defined from array A068763.
%F A068767 a(n+1) = 5*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F A068767 G.f.: (1-sqrt(1-20*x*(1-4*x)))/(10*x).
%F A068767 (n+1)*a(n) = 80*(2-n)*a(n-2) + 10*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F A068767 a(n) ~ sqrt(10+10*sqrt(5)) * (10+2*sqrt(5))^n / (10*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%F A068767 Equivalently, a(n) ~ 2^(2*n) * 5^((n-1)/2) * phi^(n + 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021
%t A068767 CoefficientList[Series[(1-Sqrt[1-20*x*(1-4*x)])/(10*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y A068767 Cf. A000108, A068764-A068766, A068768-A068772, A025227-A025230.
%K A068767 nonn,easy
%O A068767 0,3
%A A068767 _Wolfdieter Lang_, Mar 04 2002