A210591 G.f. satisfies: A(x) = x + (A(x) - A(x)^2) * A(A(x)) where A(0)=0.

1, 1, 2, 6, 24, 116, 636, 3823, 24729, 170187, 1236536, 9431875, 75202833, 624669675, 5390981898, 48229922730, 446459295023, 4269397134669, 42117578374712, 428090022559608, 4478159536973989, 48163581285504612, 532096844251876645, 6033134642314812383

Offset: 1

Author

Paul D. Hanna, Mar 23 2012

Keywords

• nonn

Comments

Compare g.f. to a g.f. C(x) of the Catalan numbers (A000108):

C(x) = x + C(x)*C(C(x)) - C(x)*C(C(x))^2 where C(x) = (1-sqrt(1-4*x))/2.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 116*x^6 + 636*x^7 + 3823*x^8 +...
Related expansions.
A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 106*x^5 + 562*x^6 + 3316*x^7 +...
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 64*x^6 + 304*x^7 + 1636*x^8 +...
A(x) - A(x)^2 = x + x^4 + 8*x^5 + 52*x^6 + 332*x^7 + 2187*x^8 +...
The series reversion of g.f. A(x) begins:
x - (x-x^2)*A(x) = x - x^2 - x^4 - 4*x^5 - 18*x^6 - 92*x^7 - 520*x^8 -...
		

Crossrefs

Cf. A190761.

Programs

PARI

{a(n)=local(A=x+x^2);for(i=1,n,A=serreverse(x-(x-x^2)*A +x*O(x^n)));polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. satisfies: A( x - (x-x^2)*A(x) ) = x.

G.f. satisfies: A( C(x) - x*A(C(x)) ) = C(x) where C(x) = (1-sqrt(1-4*x))/2 is a g.f. of the Catalan numbers.