A120977 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 * A(x*A(x)^5)^5.

1, 1, 10, 170, 3745, 96960, 2814752, 89221360, 3037327145, 109825686370, 4185287088735, 167139924222426, 6964610755602495, 301800832258018835, 13564159649547824735, 630916661388096564620, 30316241123672291911875

Offset: 0

Author

Paul D. Hanna, Jul 20 2006

Keywords

• nonn

Crossrefs

Cf. A120976; variants: A110447, A120971, A120973, A120975.

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)^5))[ #G]); A=Vec(((Ser(G)-1)/x)^(1/5));A[n+1]}

PARI

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*a(n-j, 5*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)^5) and A(x/G(x)^5) = G(x), where G(x) is the g.f. of A120976 and satisfies G(x/G(x)^5) = 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(5*n+k,j)/(5*n+k) * a(n-j,5*j). (End)