A359722 a(n) = A359720(3*n+1,2*n) for n >= 0.

1, 9, 54, 269, 1254, 5642, 24828, 107613, 461318, 1961102, 8282196, 34792914, 145527004, 606473844, 2519619640, 10440010845, 43158028230, 178049440230, 733229991780, 3014712182790, 12377406450420, 50751988872780, 207859022097480, 850399040956530, 3475797671194524

Offset: 0

Author

Paul D. Hanna, Jan 14 2023

Keywords

• nonn

Comments

The g.f. of A359720, G(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} A359720(n,k)*x^n*y^k, satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * G(x,y)^n.

A359720(3*n,2*n) = binomial(2*n+1,n)/(2*n+1) = A000108(n), n >= 0.

A359720(3*n+2,2*n+1) = binomial(2*n+1,n+1) + binomial(2*n+2,n), for n >= 0.

Crossrefs

Cf. A359720, A000108.

Programs

PARI

/* a(n) = A359720(3*n+1,2*n) */
{a(n) = my(A=[1]); for(i=1, 3*n+1, A=concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^m * x^m * (y + x^m +x*O(x^#A) )^m * Ser(A)^m ), #A-1) );
polcoeff( polcoeff(Ser(A), 3*n+1,x), 2*n,y)}
for(n=0, 30, print1(a(n), ", "))