A162661 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x)^2 ).

1, 1, 2, 7, 33, 189, 1249, 9237, 74972, 659042, 6215154, 62435805, 664459091, 7458334388, 87979090059, 1087309348481, 14041705640439, 189050930463638, 2648140182064473, 38521885088392896, 580970615943277573

Offset: 0

Author

Paul D. Hanna, Jul 09 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 189*x^5 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 84*x^4 + 472*x^5 +...
A(x*A(x)^2) = 1 + x + 4*x^2 + 20*x^3 + 121*x^4 + 838*x^5 +...
log(A(x)) = x + 3/2*x^2 + 16/3*x^3 + 103/4*x^4 + 756/5*x^5 +...
		

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,x*A^2+O(x^n)));polcoeff(A,n)}

PARI

{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(2*n-k+m,k)/(2*n-k+m)*a(n-k,k))))}

PARI

/* log(A(x)) = Sum_{n>=0} L(n)*x^n/n where: */
{L(n)=if(n<1,0,n*sum(k=1,n,binomial(2*n-k,k)/(2*n-k)*a(n-k,k)))}

Formula

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} m*C(2n-k+m,k)/(2n-k+m) * a(n-k,k).

...

Also, if log(A(x)) = Sum_{n>=0} L(n)*x^n/n, then

L(n) = n*Sum_{k=1..n} C(2n-k,k)/(2n-k) * a(n-k,k).

...