A213252 G.f. satisfies: A(x) = 1 + x/A(-x)^2.

1, 1, 2, -1, -10, 7, 88, -68, -946, 767, 11298, -9425, -144024, 122436, 1919440, -1653776, -26419778, 22992655, 372670246, -326863667, -5358911450, 4729547023, 78264621664, -69424933968, -1157715304760, 1031309398852, 17309542787288, -15474833826028

Offset: 0

Author

Paul D. Hanna, Jun 07 2012

Keywords

• sign

Examples

			G.f.: A(x) = 1 + x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...
where
x/A(-x)^2 = x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 2*x^3 - 18*x^4 - 10*x^5 + 151*x^6 + 88*x^7 +...
The g.f. G(x) of A006319 begins:
G(x) = 1 + x + 4*x^2 + 16*x^3 + 68*x^4 + 304*x^5 + 1412*x^6 + 6752*x^7 +...
where G(x) = A(x*G(x)^2) and G(x/A(x)^2) = A(x);
also, G(x) = F(x/(1-x)^2) where F(x) = 1 + x*F(x)^2 is g.f. of A000108:
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

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

Programs

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)}
for(n=0,40,print1(a(n),", "))

Formula

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).

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).

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