A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292

Offset: 0

Author

Seiichi Manyama, Jul 26 2023

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1175
• Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Crossrefs

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

Programs

Maple

A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023

Mathematica

Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)

PARI

a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).

D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023