A228712 - OEIS

%I #19 Sep 08 2013 16:04:09

%S 1,3,72,2307,86295,3513477,151235361,6768437853,311788291023,

%T 14685531568689,704028657330720,34239755370728001,1685178804762196176,

%U 83776625650642935108,4200738946110487797030,212201486734654901466543,10789009182188638106874636,551682346017956870539952958

%N G.f. A(x) satisfies: 1/A(x)^4 + 16*x*A(x)^4 = 1/A(x^2)^2 + 4*x*A(x^2)^2.

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

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

%F (2) 1/A(x)^4 + 16*x*A(x)^4 = F(x)^4,

%F (3) 1/A(x^4) + 2*x*A(x^4) = sqrt(F(x)^8 - 4*x),

%F (4) A(x) = ( (F(x)^4 - sqrt(F(x)^8 - 64*x)) / (32*x) )^(1/4),

%F (5) A(x^2) = ( (F(x)^4 - sqrt(F(x)^8 - 16*x)) / (8*x) )^(1/2),

%F where F(x) = (F(x^2)^4 + 8*x)^(1/8) is the g.f. of A223026.

%e G.f.: A(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...

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

%e 1/A(x)^4 + 16*x*A(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 +...

%e 1/A(x^2)^2 + 4*x*A(x^2)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 +...

%e Related expansions.

%e A(x)^2 = 1 + 6*x + 153*x^2 + 5046*x^3 + 191616*x^4 + 7876932*x^5 +...

%e A(x)^4 = 1 + 12*x + 342*x^2 + 11928*x^3 + 467193*x^4 + 19597332*x^5 +...

%e 1/A(x) = 1 - 3*x - 63*x^2 - 1902*x^3 - 69132*x^4 - 2764911*x^5 +...

%e 1/A(x)^2 = 1 - 6*x - 117*x^2 - 3426*x^3 - 122883*x^4 - 4875378*x^5 +...

%e The g.f. of A223026 begins:

%e F(x) = 1 + x - 3*x^2 + 14*x^3 - 76*x^4 + 441*x^5 - 2678*x^6 +...

%e where F(x)^8 = F(x^2)^4 + 8*x:

%e F(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 +...

%e F(x)^8 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1/(1/subst(A,x,x^2)^2 + 4*x*subst(A,x,x^2)^2 - 16*x*A^4 +x*O(x^n))^(1/4));polcoeff(A,n)}

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

%Y Cf. A223026.

%Y Cf. variants: A187814, A228928.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 30 2013