A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1,k) * binomial(4*n-4*k,n-k-1).
0, 1, 13, 142, 1464, 14689, 145154, 1420812, 13818784, 133793940, 1291073809, 12426782294, 119371355672, 1144851458526, 10965655515588, 104919037771224, 1002960800712720, 9580390527192940, 91453374122574372, 872513477065735768, 8320168165323802464, 79305962393873976417
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n-1, binomial(4*k+1, k)*binomial(4*n-4*k, n-k-1));
Formula
G.f.: g^3 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g)^2 * (1-4*g)^2) where g*(1-g)^3 = x.
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.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+2,k).
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