A228711 G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 8*x.

1, 2, -5, 22, -115, 646, -3822, 23496, -148368, 955822, -6256273, 41480668, -277954706, 1879118354, -12800031737, 87758481546, -605091552753, 4192829686338, -29180958305391, 203887504096188, -1429568781831693, 10055261467844862, -70929518958227340

Offset: 0

Author

Paul D. Hanna, Aug 30 2013

Keywords

• sign

Examples

			G.f.: A(x) = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +...
where A(x)^4 = A(x^2)^2 + 8*x as demonstrated by:
A(x)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 +...
A(x)^4 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...
The g.f. of A228712 begins:
G(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...
and satisfies: sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2) = A(x).
		

Crossrefs

Cf. A228712, A107086, A223026.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+8*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

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

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

where G(x) is the g.f. of A228712.

Self-convolution of A223026.