A209199 - OEIS

%I #11 Mar 13 2014 07:15:15

%S 1,1,1,3,2,5,10,21,30,76,114,257,448,1052,1706,4093,6928,16284,28266,

%T 67580,116288,278582,488152,1168105,2060388,4959066,8772450,21133812,

%U 37675236,90901086,162659624,393382077,706479172,1710430178,3084264618,7477512244,13522121028

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

%H Paul D. Hanna, <a href="/A209199/b209199.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. satisfies: 1 - x^4 - y + x^2*y + x^3*y^2 - (x*y^2)/(x^2 - y - x*y^2) = 0, where y = A(x). - _Vaclav Kotesovec_, Mar 13 2014

%F a(n) ~ (C(r,s1) - (-1)^n*C(-r,s2)) / (sqrt(Pi) * n^(3/2) * r^n), where {r1 = r = 0.45889975689289..., s1 = 3.7914195980097...} and {r2 = -r, s2 = 0.3725313335801...} are roots of the system of equations r^2*(1 + 2*r*s) = 1 + (2*r*s)/(r^2 - s - r*s^2) + (r*s^2*(1 + 2*r*s))/(-r^2 + s + r*s^2)^2, 1 + r^2*s + r^3*s^2 = r^4 + s + (r*s^2)/(r^2 - s - r*s^2), and C(r,s) = sqrt((r*s^2 - r^2 + s)*(4*r^7 - 11*r^6*s^2 - s^3 - 2*r*s^3 - 3*r^4*s^3*(s^3-6) + 10*r^5*s*(s^3-1) - 8*r^3*s^2*(s^3-1) - r^2*s^2*(7*s^2+1)) / (4*r*(r^7 - 3*r^6*s^2 + s^3 - r*s^3 - r^4*s^3*(s^3-6) + 3*r^5*s*(s^3-1) - 3*r^2*s^2*(s^2+1) + r^3*(3*s^2 - 3*s^5 - 1)))), C(r,s1) = 4.083478805997458527..., C(-r,s2) = 0.26836221180354127... - _Vaclav Kotesovec_, Mar 13 2014

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 21*x^7 +...

%e Related series:

%e A(x)*A(-x) = 1 + x^2 - x^4 + 5*x^6 + 12*x^8 + 25*x^10 + 164*x^12 +...

%e A(x)/A(-x) = 1 + 2*x + 2*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 30*x^6 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,-x)+x^2*A/subst(A,x,-x+x*O(x^n)));polcoeff(A,n)}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A208889, A208887.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 08 2012