A386896 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(3*n-k,n-k).

1, 9, 125, 1932, 31365, 523809, 8910356, 153544680, 2671398309, 46822319115, 825501663525, 14623742203200, 260088366645900, 4641248247561324, 83059406374007720, 1490097583932329232, 26790218420643034533, 482571492068274975135, 8707190579448431827991

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(2*n+1).

a(n) = [x^n] 1/((1-x) * (1-2*x))^(2*n+1).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(3*n-k,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(3*n-k,n-k).

a(n) = binomial(3*n, n)*hypergeom([-1-5*n, -n], [-3*n], -1). - Stefano Spezia, Aug 07 2025

D-finite with recurrence 202*n*(n-1)*(2*n-1)*(2*n-3)*a(n) -3*(n-1)*(2*n-3) *(14093*n^2-15245*n+5226)*a(n-1) +4*(355081*n^4 -1597876*n^3 +2789549*n^2 -2405270*n+926160)*a(n-2) -3840*(5*n-11)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 21 2025

Recurrence (of order 2): 2*(n-1)*n*(2*n - 3)*(2*n - 1)*(41*n - 52)*a(n) = 3*(n-1)*(2*n - 3)*(5617*n^3 - 12741*n^2 + 8390*n - 1512)*a(n-1) - 20*(5*n - 8)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(41*n - 11)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Prog

(PARI) a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(3*n-k, n-k));

Crossrefs

Cf. A386812, A386895, A386897, A386898.

Sequence in context: A138438 A320646 A241709 * A085528 A192724 A078422

Adjacent sequences: A386893 A386894 A386895 * A386897 A386898 A386899

Keyword

nonn

Author

Seiichi Manyama, Aug 07 2025

Status

approved