A120566 G.f. satisfies: A(x) = A(A(x)) - x*A(A(A(x))), with A(0)=0.

1, 1, 1, 3, 7, 33, 109, 643, 2623, 17929, 85349, 652395, 3517911, 29484193, 176844781, 1605009651, 10575269935, 103033059513, 738834271605, 7676696689275, 59466011617671, 655467253898577, 5451048833933693

Offset: 1

Author

Paul D. Hanna, Jun 14 2006

Keywords

• nonn

Comments

If A(0, x) = x, A(n+1, x) = A( A(n, x)) = A(n, A(x)). Then A(n, x) = x + n*x^2 + n^2*x^3 + (n^3 + 2*n)*x^4 + (n^4 + 6*n^2)*x^5 + ... where [x^4] A(n, x) = A054602(n). - Michael Somos, Jan 22 2012

Examples

			A(x) = x + x^2 + x^3 + 3x^4 + 7x^5 + 33x^6 + 109x^7 + 643x^8 +...
A(A(x)) = x + 2x^2 + 4x^3 + 12x^4 + 40x^5 + 168x^6 + 736x^7 + 3784x^8+..
x*A(A(A(x))) = x^2 + 3x^3 + 9x^4 + 33x^5 + 135x^6 + 627x^7 + 3141x^8+...
		

Programs

PARI

{a(n)=local(A=x+x^2+x*O(x^n));if(n<1,0, for(i=1,n,A=x-subst(A,x,-x)*subst(A,x,A));polcoeff(A,n))}

Formula

G.f. satisfies: A(-A(-x)) = x ; Also: A(x) = x + A(A(x))*series_reversion(A(x)).

Since g.f. satisfies: A(A(x)) = ( x - A(x) )/A(-x), then higher order self-compositions of A(x) reduce into expressions involving A(x) and A(-x). - Paul D. Hanna, Jul 22 2006