A228712 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.

1, 3, 72, 2307, 86295, 3513477, 151235361, 6768437853, 311788291023, 14685531568689, 704028657330720, 34239755370728001, 1685178804762196176, 83776625650642935108, 4200738946110487797030, 212201486734654901466543, 10789009182188638106874636, 551682346017956870539952958

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

G.f. A(x) satisfies:

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

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

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

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

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

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

Example

G.f.: A(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^4 + 16*x*A(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 +...
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 +...
Related expansions.
A(x)^2 = 1 + 6*x + 153*x^2 + 5046*x^3 + 191616*x^4 + 7876932*x^5 +...
A(x)^4 = 1 + 12*x + 342*x^2 + 11928*x^3 + 467193*x^4 + 19597332*x^5 +...
1/A(x) = 1 - 3*x - 63*x^2 - 1902*x^3 - 69132*x^4 - 2764911*x^5 +...
1/A(x)^2 = 1 - 6*x - 117*x^2 - 3426*x^3 - 122883*x^4 - 4875378*x^5 +...
The g.f. of A223026 begins:
F(x) = 1 + x - 3*x^2 + 14*x^3 - 76*x^4 + 441*x^5 - 2678*x^6 +...
where F(x)^8 = F(x^2)^4 + 8*x:
F(x)^4 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 +...
F(x)^8 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...

Prog

(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)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A223026.

Cf. variants: A187814, A228928.

Sequence in context: A332188 A071645 A322189 * A300967 A332721 A332747

Adjacent sequences: A228709 A228710 A228711 * A228713 A228714 A228715

Keyword

nonn

Author

Paul D. Hanna, Aug 30 2013

Status

approved