A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

1, 2, 12, 114, 1435, 22219, 406441, 8559852, 203792337, 5409449156, 158350300141, 5066765087000, 175908765569628, 6585443884172129, 264428161094825151, 11335716352419699208, 516717363793695685925, 24955728581736822645816

Offset: 1

Author

Paul D. Hanna, Jun 10 2008

Keywords

• nonn

Examples

			a(n) = the n-th coefficient of the n-th iteration of G(x):
[x] G(x) = 1, [x^2] G(G(x)) = 2, [x^3] G(G(G(x))) = 12, etc.
The initial iterations (n=1..7) of G(x) are:
n=1: x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 +...
n=2: x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + 2982*x^7 +...
n=3: x + 3*x^2 + 12*x^3 + 57*x^4 + 305*x^5 + 1787*x^6 + 11269*x^7 +...
n=4: x + 4*x^2 + 20*x^3 + 114*x^4 + 712*x^5 + 4772*x^6 + 33896*x^7 +...
n=5: x + 5*x^2 + 30*x^3 + 200*x^4 + 1435*x^5 + 10900*x^6 + 86799*x^7+...
n=6: x + 6*x^2 + 42*x^3 + 321*x^4 + 2608*x^5 + 22219*x^6 + 196910*x^7+...
n=7: x + 7*x^2 + 56*x^3 + 483*x^4 + 4389*x^5 + 41531*x^6 + 406441*x^7+...
Notice the main diagonal of the table formed from these coefficients.
		

Crossrefs

Cf. A030266, A128325, A128329.

Programs

PARI

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