A182224 G.f. satisfies: A(x) = 1 + x*A(x) * A(A(x) - 1).

1, 1, 2, 7, 35, 220, 1622, 13480, 123212, 1218694, 12898292, 144890911, 1717072304, 21367220392, 278174617499, 3777454890226, 53372573914742, 783004637781214, 11905653060557285, 187325244287570431, 3045651218248945454, 51103551998888439679, 883901254149820933025

Offset: 0

Author

Paul D. Hanna, Apr 19 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
such that A(x) = 1 + x*A(x)*A(A(x)-1) where:
A(A(x)-1) = 1 + x + 4*x^2 + 22*x^3 + 148*x^4 + 1147*x^5 + 9901*x^6 +...
Let G(x) satisfy G(x/A(x)) = x, then G(x)/x = g.f. of A088717, where
G(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
A(G(x)) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
A(x/(1+G(x))) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
G(x*A(x)) = x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
		

Crossrefs

Cf. A088717.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*A*subst(A,x,A-1)+x*O(x^n));polcoeff(A,n)}

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+subst(serreverse(x/A),x,x*A)+x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

Given g.f. A(x), define G(x) by G(x)/x = g.f. of A088717, then G(x) satisfies:

(1) G(x) = x + x*G(G(x)^2/x),

(2) G(x) = Series_Reversion(x/A(x)),

(3) G(x) = x/Series_Reversion(x*A(x)) - 1,

(4) G(x) = x*A(G(x)),

(5) G(x) = A(x/(1+G(x))) - 1,

(6) A(x) = 1 + G(x*A(x)).