A335647 a(n) = binomial(4*n+1,n+1).

1, 10, 84, 715, 6188, 54264, 480700, 4292145, 38567100, 348330136, 3159461968, 28760021745, 262596783764, 2403979904200, 22057981462440, 202802465047245, 1867897112363100, 17231414395464984, 159186450151978480, 1472474663905800940, 13636219405675529520

Offset: 0

Author

Vladimir Kruchinin, Jun 15 2020

Keywords

• nonn

Crossrefs

Cf. A002293.

Programs

Maxima

a(n):=sum(binomial(n+1,k)*binomial(3*n,k),k,0,n+1);

PARI

a(n) = binomial(4*n+1, n+1); \\ Michel Marcus, Jun 15 2020

Formula

G.f.: A(x) = x*B'(x)/B(x)+x*(1/x-1/B(x))', where B(x) = x*(1+B(x))^4 = A002293(x)-1.

a(n) = Sum_{k=0..n+1} C(n+1,k)*C(3*n,k).

From Karol A. Penson, Mar 07 2025: (Start)

G.f.: h(z) = 3*(hypergeom([-1/2,-1/4,1/4],[-2/3,-1/3],256*z/27)-1)/(4*z) satisfies

z^2 + 41*z + 27 + (73*z^2 + 310*z - 27)*h(z) + z*(32*z^2 + 795*z - 81)*h(z)^2 + 3*z^2*(256*z - 27)*h(z)^3 + z^3*(256*z - 27)*h(z)^4 = 0. (End)