%I #13 Jun 09 2012 15:00:34

%S 1,1,2,-1,-10,7,88,-68,-946,767,11298,-9425,-144024,122436,1919440,

%T -1653776,-26419778,22992655,372670246,-326863667,-5358911450,

%U 4729547023,78264621664,-69424933968,-1157715304760,1031309398852,17309542787288,-15474833826028

%N G.f. satisfies: A(x) = 1 + x/A(-x)^2.

%H Paul D. Hanna, <a href="/A213252/b213252.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. satisfies: A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A006319 (royal paths in a lattice).

%F G.f. satisfies: A(x) = sqrt( x/Series_Reversion( x*C(x/(1-x)^2)^2 ) ) where C(x) = 1 + x*C(x)^2 = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

%F G.f. satisfies: A(x) = A(x)*A(-x) + x/A(x).

%e G.f.: A(x) = 1 + x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...

%e where

%e x/A(-x)^2 = x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...

%e A(x)^2 = 1 + 2*x + 5*x^2 + 2*x^3 - 18*x^4 - 10*x^5 + 151*x^6 + 88*x^7 +...

%e The g.f. G(x) of A006319 begins:

%e G(x) = 1 + x + 4*x^2 + 16*x^3 + 68*x^4 + 304*x^5 + 1412*x^6 + 6752*x^7 +...

%e where G(x) = A(x*G(x)^2) and G(x/A(x)^2) = A(x);

%e also, G(x) = F(x/(1-x)^2) where F(x) = 1 + x*F(x)^2 is g.f. of A000108:

%e F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

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

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A006319, A213281, A213335, A143045; A000108.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jun 07 2012