A187814 G.f. A(x) satisfies: 1/A(x)^2 + 4*x*A(x)^2 = 1/A(x^2) + 2*x*A(x^2).

1, 1, 6, 41, 334, 2901, 26651, 253709, 2483395, 24829132, 252506507, 2603798287, 27161758393, 286118173600, 3039211373800, 32517513415886, 350122302629869, 3790909121211262, 41249405668333107, 450832515809731316, 4947009705400704588, 54479711308604703264, 601933495810972446631

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies:

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

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

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

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

where F(x) = (F(x^2)^2 + 4*x)^(1/4) is the g.f. of A107086.

Example

G.f.: A(x) = 1 + x + 6*x^2 + 41*x^3 + 334*x^4 + 2901*x^5 + 26651*x^6 +...
such that A(x) satisfies the identity illustrated by:
1/A(x)^2 + 4*x*A(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
1/A(x^2) + 2*x*A(x^2) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 13*x^2 + 94*x^3 + 786*x^4 + 6962*x^5 + 64793*x^6 +...
A(x)^4 = 1 + 4*x + 30*x^2 + 240*x^3 + 2117*x^4 + 19512*x^5 + 186706*x^6 +...
1/A(x) = 1 - x - 5*x^2 - 30*x^3 - 233*x^4 - 1949*x^5 - 17503*x^6 +...
1/A(x)^2 = 1 - 2*x - 9*x^2 - 50*x^3 - 381*x^4 - 3132*x^5 - 27878*x^6 +...
The g.f. of A107086 begins:
F(x) = 1 + x - x^2 + 2*x^3 - 5*x^4 + 13*x^5 - 35*x^6 + 99*x^7 - 289*x^8 +...
where F(x)^4 = F(x^2)^2 + 4*x:
F(x)^2 = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 +...
F(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 +...

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*x*subst(A, x, x^2) - 4*x*A^2 +x*O(x^n))^(1/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A107086.

Cf. variants: A228712, A228928.

Sequence in context: A345189 A083430 A005011 * A009122 A184140 A317410

Adjacent sequences: A187811 A187812 A187813 * A187815 A187816 A187817

Keyword

nonn

Author

Paul D. Hanna, Aug 30 2013

Status

approved