A214761 - OEIS

%I #6 Aug 22 2024 16:44:19

%S 1,2,4,12,40,128,416,1344,4224,12928,38016,104832,260096,512256,

%T 329728,-4140032,-33444864,-184423424,-883798016,-3935711232,

%U -16759001088,-69266997248,-280327684096,-1116872122368,-4394989174784,-17112512544768,-65974620848128

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

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

%C 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).

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

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

%e 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 +...

%e Related expansions:

%e 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 +...

%o (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)}

%o for(n=0,31,print1(a(n),", "))

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

%K sign

%O 0,2

%A _Paul D. Hanna_, Jul 29 2012