%I #17 Sep 26 2012 17:55:17

%S 1,1,1,2,7,26,88,389,1617,7808,34783,184426,878705,4960662,25295649,

%T 152048803,820978097,5254469132,30147771222,204763245407,

%U 1249116889562,9012614482274,58336358284152,446435253834922,3064458931156669,24788742473819564,179927874744752672

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * ( (A(x)^n + A(-x)^n)/2 )^n.

%F G.f.: Sum_{n>=0} (x/2)^n * A(x)^(n^2) * Sum_{k=0..n} binomial(n,k) * (A(-x)/A(x))^(n*k).

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 88*x^6 + 389*x^7 +...

%e where

%e A(x) = 1 + x*(A(x)+A(-x))/2 + x^2*(A(x)^2+A(-x)^2)^2/2^2 + x^3*(A(x)^3+A(-x)^3)^3/2^3 + x^4*(A(x)^4+A(-x)^4)^4/2^4 + x^5*(A(x)^5+A(-x)^5)^5/2^5 +...

%e Related expansions:

%e (A(x)+A(-x))/2 = 1 + x^2 + 7*x^4 + 88*x^6 + 1617*x^8 + 34783*x^10 +...

%e (A(x)^2+A(-x)^2)^2/2^2 = 1 + 6*x^2 + 47*x^4 + 606*x^6 + 10519*x^8 +...

%e (A(x)^3+A(-x)^3)^3/2^3 = 1 + 18*x^2 + 225*x^4 + 3144*x^6 + 53190*x^8 +...

%e (A(x)^4+A(-x)^4)^4/2^4 = 1 + 40*x^2 + 884*x^4 + 16208*x^6 + 298066*x^8 +...

%e (A(x)^5+A(-x)^5)^5/2^5 = 1 + 75*x^2 + 2850*x^4 + 77525*x^6 + 1802600*x^8 +...

%e (A(x)^6+A(-x)^6)^6/2^6 = 1 + 126*x^2 + 7767*x^4 + 321174*x^6 + 10371699*x^8 +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*((A^m+subst(A^m, x, -x))/2)^m)); polcoeff(A, n)}

%o for(n=0, 21, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, (x/2)^m*A^(m^2)*sum(k=0,m,binomial(m,k)*(subst(A, x, -x)/A)^(m*k)))); polcoeff(A, n)}

%Y Cf. A217041.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 21 2012