A210591 - OEIS

%I #14 Apr 20 2012 10:37:03

%S 1,1,2,6,24,116,636,3823,24729,170187,1236536,9431875,75202833,

%T 624669675,5390981898,48229922730,446459295023,4269397134669,

%U 42117578374712,428090022559608,4478159536973989,48163581285504612,532096844251876645,6033134642314812383

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

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

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

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

%F 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.

%e 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 +...

%e Related expansions.

%e A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 106*x^5 + 562*x^6 + 3316*x^7 +...

%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 64*x^6 + 304*x^7 + 1636*x^8 +...

%e A(x) - A(x)^2 = x + x^4 + 8*x^5 + 52*x^6 + 332*x^7 + 2187*x^8 +...

%e The series reversion of g.f. A(x) begins:

%e x - (x-x^2)*A(x) = x - x^2 - x^4 - 4*x^5 - 18*x^6 - 92*x^7 - 520*x^8 -...

%o (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)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A190761.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 23 2012