A371813 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n-k-1,n-k).
1, 1, 7, 40, 239, 1461, 9076, 57044, 361711, 2309467, 14827487, 95630272, 619111172, 4021011580, 26187682024, 170960159100, 1118406332655, 7330011083079, 48119501497909, 316354663355384, 2082573599282359, 13726029056757029, 90565080767425744
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, (-1)^k*binomial(3*n-k-1, n-k));
Formula
a(n) = [x^n] 1/((1+x) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, -n], [1-3*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 8*n*(2*n - 1)*(28*n^2 - 87*n + 67)*a(n) = 2*(1456*n^4 - 6008*n^3 + 8593*n^2 - 4949*n + 960)*a(n-1) + 3*(3*n - 5)*(3*n - 4)*(28*n^2 - 31*n + 8)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n+2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(-3+4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025