A214761 G.f. satisfies: A(x) = 1/A(-x*A(x)).

1, 2, 4, 12, 40, 128, 416, 1344, 4224, 12928, 38016, 104832, 260096, 512256, 329728, -4140032, -33444864, -184423424, -883798016, -3935711232, -16759001088, -69266997248, -280327684096, -1116872122368, -4394989174784, -17112512544768, -65974620848128

Offset: 0

Author

Paul D. Hanna, Jul 29 2012

Keywords

• sign

Comments

Compare g.f. to: G(x) = 1/G(-x*G(x)) when G(x) = 1/(1-x).

An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)); for example, (*) is satisfied by G(x) = 1/(1-m*x).

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 128*x^5 + 416*x^6 + 1344*x^7 +...
Related expansions:
1/A(x) = A(-x*A(x)) = 1 - 2*x - 4*x^3 - 8*x^4 - 16*x^5 - 48*x^6 - 96*x^7 - 128*x^8 + 64*x^9 + 2048*x^10 +...
		

Crossrefs

Cf. A214762, A214763, A214764, A214765, A214766, A214767, A214768, A214769.

Programs

PARI

{a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^1+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

Formula

The g.f. of this sequence is the limit of the recurrence:

(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)))/2 starting at G_0(x) = 1+2*x.