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 A337291 #30 Feb 28 2025 06:18:57
%S A337291 4,12,60,364,2448,17556,131560,1017900,8069424,65204656,535070172,
%T A337291 4446927732,37353738800,316621743480,2704784196240,23263187479980,
%U A337291 201275443944432,1750651680235920,15298438066553776,134252511729576240,1182622941581590080
%N A337291 a(n) = 3*binomial(4*n,n)/(4*n-1).
%C A337291 a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/3 are the path's endpoints. - _Lucas A. Brown_, Aug 21 2020
%H A337291 R. J. Mathar, <a href="https://vixra.org/abs/2502.0097">The Eggenberger-Polya urn process: Probabilities of revisited ball ratios</a>, vixra:2502 (2025)
%F A337291 a(n) = 4*A006632(n).
%F A337291 G.f.: 4*x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
%F A337291 D-finite with recurrence 3*n*(3*n-1)*(3*n-2)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-1)*a(n-1)=0, a(0)=1. - _R. J. Mathar_, Jan 26 2025
%t A337291 Array[3 Binomial[4 #, #]/(4 # - 1) &, 21] (* _Michael De Vlieger_, Aug 21 2020 *)
%o A337291 (PARI) a(n) = {3*binomial(4*n,n)/(4*n-1)} \\ _Andrew Howroyd_, Aug 21 2020
%Y A337291 Cf. A006632, A337292.
%K A337291 nonn,easy
%O A337291 1,1
%A A337291 _Lucas A. Brown_, Aug 21 2020