%I #3 Sep 08 2013 15:47:08

%S 1,7,672,91147,14486409,2516759469,463051052653,88674496050245,

%T 17490154693966234,3528922457876864195,724934544034900295558,

%U 151110852750623222310189,31881833636363854856989129,6795336519252277650628254056,1461001691259055273207790036665

%N G.f. A(x) satisfies: 1/A(x)^8 + 64*x*A(x)^8 = 1/A(x^2)^4 + 8*x*A(x^2)^4.

%F G.f. A(x) satisfies:

%F (1) 1/A(x)^8 + 64*x*A(x)^8 = F(x)^8,

%F (2) 1/A(x^2)^4 + 8*x*A(x^2)^4 = F(x)^8,

%F (3) A(x) = ( (F(x)^8 - sqrt(F(x)^16 - 256*x)) / (128*x) )^(1/8),

%F (4) A(x^2) = ( (F(x)^8 - sqrt(F(x)^16 - 32*x)) / (16*x) )^(1/4),

%F where F(x) = (F(x^2)^8 + 16*x)^(1/16) is the g.f. of A228927.

%e G.f.: A(x) = 1 + 7*x + 672*x^2 + 91147*x^3 + 14486409*x^4 +...

%e such that A(x) satisfies the identity illustrated by:

%e 1/A(x)^8 + 64*x*A(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 +...

%e 1/A(x^2)^4 + 8*x*A(x^2)^4 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 +...

%e Related expansions.

%e A(x)^4 = 1 + 28*x + 2982*x^2 + 422408*x^3 + 68709025*x^4 + 12111355116*x^5 +...

%e A(x)^8 = 1 + 56*x + 6748*x^2 + 1011808*x^3 + 169965222*x^4 + 30589656944*x^5 +...

%e 1/A(x)^4 = 1 - 28*x - 2198*x^2 - 277368*x^3 - 42560861*x^4 - 7240234148*x^5 +...

%e 1/A(x)^8 = 1 - 56*x - 3612*x^2 - 431648*x^3 - 64757910*x^4 - 10877750352*x^5 +...

%e The g.f. of A228927 satisfies F(x) = (F(x^2)^8 + 16*x)^(1/16) and begins:

%e F(x) = 1 + x - 7*x^2 + 70*x^3 - 798*x^4 + 9737*x^5 - 124124*x^6 + 1631041*x^7 +...

%e where F(x)^16 = F(x^2)^8 + 16*x:

%e F(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 - 277368*x^6 +...

%e F(x)^16 = 1 + 16*x + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1/subst(A, x, x^2)^4 + 8*x*subst(A, x, x^2)^4 - 64*x*A^8 +x*O(x^n))^(1/8)); polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A228927.

%Y Cf. variants: A187814, A228712.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 08 2013