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 A071736 #17 Aug 04 2025 02:27:13
%S A071736 1,3,9,29,96,324,1111,3861,13572,48178,172482,622098,2258416,8246190,
%T A071736 30264435,111585765,413126460,1535267250,5724840990,21413721510,
%U A071736 80326153440,302105210160,1138957917318,4303550907234,16294686579016
%N A071736 Expansion of (1+x^3*C^3)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%C A071736 a(n) = number of Dyck (n+3)-paths whose initial ascent has length divisible by 3. For example, UUUUDDUDDD has initial ascent of length 4 and a(1) counts UUUDUDDD, UUUDDUDD, UUUDDDUD. - _David Callan_, Jul 25 2005
%H A071736 Vincenzo Librandi, <a href="/A071736/b071736.txt">Table of n, a(n) for n = 0..200</a>
%F A071736 a(n) ~ 15*4^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F A071736 a(n) = 3*binomial(2*n,n)*(5*n^2+3*n+4)/((n+1)*(n+2)*(n+3)) for n>0. - _Tani Akinari_, Aug 03 2025
%t A071736 CoefficientList[Series[(1 + x^3 ((1 - (1 - 4 x)^(1/2))/(2 x))^3) ((1 - (1 - 4 x)^(1/2))/(2 x))^3, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o A071736 (Maxima) a(n):=if n=0 then 1 else 3*binomial(2*n,n)*(5*n^2+3*n+4)/((n+1)*(n+2)*(n+3)); /* _Tani Akinari_, Aug 03 2025 */
%K A071736 nonn
%O A071736 0,2
%A A071736 _N. J. A. Sloane_, Jun 06 2002