OFFSET
0,2
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).
G.f.: h(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)
a(n) ~ 2^(8*n+5/2) / (3^(3*n+1/2) * sqrt(Pi*n)). - Amiram Eldar, Sep 21 2025
MATHEMATICA
a[n_] := Binomial[4*n+1, n+1]; Array[a, 20, 0] (* Amiram Eldar, Sep 21 2025 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Kruchinin, Jun 15 2020
STATUS
approved