A173139 - OEIS

%I #2 Mar 30 2012 18:37:21

%S 1,1,2,4,11,31,97,309,1026,3466,11964,41856,148373,531233,1919313,

%T 6986745,25604367,94379887,349702606,1301729084,4865680876,

%U 18255350676,68724316775,259521249065,982796892300,3731493081316,14201727734640

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

%F G.f. satisfies: A(x/(x + A(x^2))) = x + A(x^2).

%F G.f. satisfies: A(x) = (1/x)*Series_Reversion(x/(x + A(x^2))).

%F Given e.g.f. E(x), then E(x)/exp(x) is an even function.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 97*x^6 +...

%e Let G(x) = x + A(x^2):

%e G(x) = 1 + x + x^2 + 2*x^4 + 4*x^6 + 11*x^8 + 31*x^10 + 97*x^12 +...

%e then A(x) = G(x*A(x)) and G(x) = A(x/G(x)).

%e Given e.g.f. E(x), then E(x)/exp(x) is the even function:

%e E(x)/exp(x) = 1 + x^2/2! + 4*x^4/4! + 21*x^6/6! + 129*x^8/8! + 863*x^10/10! + 6109*x^12/12! +...

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 10 2010