A217138 G.f. A(x) satisfies A(x) = 1 + x*A(x)^2*(A(x) + A(-x))/2.

1, 1, 2, 7, 22, 94, 340, 1579, 6118, 29746, 120060, 600934, 2492028, 12725756, 53798888, 278786739, 1195684230, 6265816042, 27175425004, 143671870034, 628705751828, 3347680236132, 14756641134872, 79039468217086, 350529497005532, 1886818634445044, 8410852483002200

Offset: 0

Author

Paul D. Hanna, Sep 27 2012

Keywords

• nonn

Comments

Compare to: G(x) = 1 + x*G(x)*(G(x) + G(-x))/2, which is the g.f. of A047749.

The radius of convergence r of g.f. A(x) is

r = 0.192450089729875254836382926833985818549200... with

A(r) = (3 - sqrt(9-6*sqrt(2)))/sqrt(2) = 1.614014407382354328773...

A(-r) = (sqrt(9+6*sqrt(2)) - 3)/sqrt(2) = 0.835475335400823769423...

where y = A(r) and y = A(-r) solves y^4 = 18*(1-y)^2.

In closed form, r = 1/(3*sqrt(3)). - Vaclav Kotesovec, Feb 17 2014

Examples

			A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 94*x^5 + 340*x^6 + 1579*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 62*x^4 + 260*x^5 + 1005*x^6 + 4522*x^7 +...
(A(x) + A(-x))/2 = 1 + 2*x^2 + 22*x^4 + 340*x^6 + 6118*x^8 +...
		

Crossrefs

Cf. A047749.

Programs

PARI

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

Formula

G.f. A(x) satisfies: 2*(A(x)-1)^3 + x*(A(x)-1)*(2-A(x))*A(x)^3 - x^2*A(x)^6 = 0.

The formal inverse of g.f. A(x) is (x-1)*(2-x + sqrt(x^2 + 4*x - 4))/(2*x^3).

Recurrence: (n-1)*n*(n+1)*(2*n-1)*(2*n+1)*(198*n^5 - 1890*n^4 + 6915*n^3 - 12015*n^2 + 9832*n - 3060)*a(n) = 30*(n-1)*n*(2*n - 1)*(3*n - 7)*(36*n^4 - 246*n^3 + 567*n^2 - 468*n + 55)*a(n-1) + 3*(n-1)*(14256*n^9 - 193104*n^8 + 1111626*n^7 - 3526794*n^6 + 6652659*n^5 - 7420161*n^4 + 4436059*n^3 - 971921*n^2 - 90480*n - 23100)*a(n-2) - 90*(2*n - 1)*(3*n - 8)*(3*n - 7)*(3*n - 4)*(36*n^4 - 246*n^3 + 567*n^2 - 468*n + 55)*a(n-3) - 36*(n-3)*(3*n - 11)*(3*n - 10)*(3*n - 8)*(3*n - 7)*(198*n^5 - 900*n^4 + 1335*n^3 - 630*n^2 - 23*n - 20)*a(n-4). - Vaclav Kotesovec, Feb 17 2014

a(n) ~ (1+2*sqrt(2)-sqrt(3) + (-1)^n*(1-2*sqrt(2)+sqrt(3))) * 3^(3*n/2) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2014

a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+j+2*k=n-1} a(i) * a(j) * a(2*k). - Seiichi Manyama, Jul 07 2025