OFFSET
0,2
FORMULA
G.f. A(x) satisfies:
(1) 1/A(x)^8 + 64*x*A(x)^8 = F(x)^8,
(2) 1/A(x^2)^4 + 8*x*A(x^2)^4 = F(x)^8,
(3) A(x) = ( (F(x)^8 - sqrt(F(x)^16 - 256*x)) / (128*x) )^(1/8),
(4) A(x^2) = ( (F(x)^8 - sqrt(F(x)^16 - 32*x)) / (16*x) )^(1/4),
where F(x) = (F(x^2)^8 + 16*x)^(1/16) is the g.f. of A228927.
EXAMPLE
G.f.: A(x) = 1 + 7*x + 672*x^2 + 91147*x^3 + 14486409*x^4 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^8 + 64*x*A(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 +...
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 +...
Related expansions.
A(x)^4 = 1 + 28*x + 2982*x^2 + 422408*x^3 + 68709025*x^4 + 12111355116*x^5 +...
A(x)^8 = 1 + 56*x + 6748*x^2 + 1011808*x^3 + 169965222*x^4 + 30589656944*x^5 +...
1/A(x)^4 = 1 - 28*x - 2198*x^2 - 277368*x^3 - 42560861*x^4 - 7240234148*x^5 +...
1/A(x)^8 = 1 - 56*x - 3612*x^2 - 431648*x^3 - 64757910*x^4 - 10877750352*x^5 +...
The g.f. of A228927 satisfies F(x) = (F(x^2)^8 + 16*x)^(1/16) and begins:
F(x) = 1 + x - 7*x^2 + 70*x^3 - 798*x^4 + 9737*x^5 - 124124*x^6 + 1631041*x^7 +...
where F(x)^16 = F(x^2)^8 + 16*x:
F(x)^8 = 1 + 8*x - 28*x^2 + 224*x^3 - 2198*x^4 + 23856*x^5 - 277368*x^6 +...
F(x)^16 = 1 + 16*x + 8*x^2 - 28*x^4 + 224*x^6 - 2198*x^8 + 23856*x^10 +...
PROG
(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)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 08 2013
STATUS
approved