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 A386612 #21 Aug 10 2025 08:05:35
%S A386612 0,1,13,142,1464,14689,145154,1420812,13818784,133793940,1291073809,
%T A386612 12426782294,119371355672,1144851458526,10965655515588,
%U A386612 104919037771224,1002960800712720,9580390527192940,91453374122574372,872513477065735768,8320168165323802464,79305962393873976417
%N A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1,k) * binomial(4*n-4*k,n-k-1).
%F A386612 G.f.: g^3 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
%F A386612 G.f.: g/((1-g)^2 * (1-4*g)^2) where g*(1-g)^3 = x.
%F A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
%F A386612 a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+2,k).
%F A386612 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+2,k).
%F A386612 D-finite with recurrence 13122*n*(3*n+2)*(3*n+1)*a(n) +81*(-124803*n^3+284553*n^2-210740*n+42140)*a(n-1) +24*(7476768*n^3-29253744*n^2+37920106*n-16562575)*a(n-2) +40960*(-26344*n^3+148032*n^2-282329*n+185874)*a(n-3) +55050240*(2*n-5)*(4*n-13)*(4*n-11)*a(n-4)=0. - _R. J. Mathar_, Aug 10 2025
%o A386612 (PARI) a(n) = sum(k=0, n-1, binomial(4*k+1, k)*binomial(4*n-4*k, n-k-1));
%Y A386612 Cf. A078995, A308523, A386565, A386611.
%Y A386612 Cf. A006419, A386614, A386616, A386617.
%K A386612 nonn
%O A386612 0,3
%A A386612 _Seiichi Manyama_, Jul 27 2025