A217138 - OEIS

%I #9 Feb 17 2014 16:59:56

%S 1,1,2,7,22,94,340,1579,6118,29746,120060,600934,2492028,12725756,

%T 53798888,278786739,1195684230,6265816042,27175425004,143671870034,

%U 628705751828,3347680236132,14756641134872,79039468217086,350529497005532,1886818634445044,8410852483002200

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

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

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

%C r = 0.192450089729875254836382926833985818549200... with

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

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

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

%C In closed form, r = 1/(3*sqrt(3)). - _Vaclav Kotesovec_, Feb 17 2014

%F 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.

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

%F 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

%F 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

%e A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 94*x^5 + 340*x^6 + 1579*x^7 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 62*x^4 + 260*x^5 + 1005*x^6 + 4522*x^7 +...

%e (A(x) + A(-x))/2 = 1 + 2*x^2 + 22*x^4 + 340*x^6 + 6118*x^8 +...

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A047749.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 27 2012