A120973 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 * A(x*A(x)^3)^3.

1, 1, 6, 60, 776, 11802, 201465, 3759100, 75404151, 1608036861, 36172106112, 853346084343, 21021015647574, 538868533164995, 14336235065928966, 394957784033440194, 11246848201518516044, 330520280036501809758

Offset: 0

Author

Paul D. Hanna, Jul 20 2006

Keywords

• nonn

Crossrefs

Cf. A120972; variants: A110447, A120971, A120975, A120977.

Programs

PARI

{a(n)=local(A,G=[1,1]);for(i=1,n,G=concat(G,0); G[ #G]=-Vec(subst(Ser(G),x,x/Ser(G)^3))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/3));A[n+1]}

PARI

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+k, j)/(3*n+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

G.f. A(x) satisfies: A(x) = G(G(x)-1), A(G(x)-1) = G(A(x)-1), A(x) = G(x*A(x)^3) and A(x/G(x)^3) = G(x), where G(x) is the g.f. of A120972 and satisfies G(x/G(x)^3) = 1 + x.

From Seiichi Manyama, Mar 01 2025: (Start)

Let a(n,k) = [x^n] A(x)^k.

a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n+k,j)/(3*n+k) * a(n-j,3*j). (End)