A371678 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x)^(1/2))^2.

1, 4, 56, 1068, 23504, 561972, 14183880, 371911132, 10031990560, 276589937892, 7759696110808, 220805824681740, 6357540660485616, 184876232243020564, 5422016433851400552, 160187931368799105468, 4763038761416835095616, 142426926824923660491716

Offset: 0

Author

Seiichi Manyama, Apr 02 2024

Keywords

• nonn

Links

• Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Crossrefs

Cf. A006319, A032349, A371675. A371676, A371677.

Cf. A371700.

Programs

PARI

a(n, r=2, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

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

G.f.: B(x)^2 where B(x) is the g.f. of A371700.

a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+2,n)/(6*n+k+2).