The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2. 5
 1, 1, 4, 8, 28, 80, 308, 984, 3980, 13472, 56164, 197032, 838396, 3013872, 13015188, 47624568, 207971436, 771336512, 3397886660, 12736715592, 56502898140, 213618833808, 953139545076, 3629043226392, 16270547827020, 62317467147744 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Specific values: A(2/9) = 17/9 and A(-2/9) = 17/18. Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462... with A(r) = (2 + sqrt(1-3*r))*(1+r^2)/(1+r) = 2.19775350... and A(-r) = (2 - sqrt(1+3*r))*(1+r^2)/(1-r) = 3*(1+r^2) - A(r) = 0.9569470... At x=r, the equation (*) (1+x^2)^2 - 2*(1+x^2)*y + (1+x)*y^2 - x*y^3 = 0, which is satisfied by y = A(x), factors out to: (y - A(r))^2 * (y - A(r)*(1+r^2)/(2*(A(r)-1-r^2))) = 0; this gives the relation: (A(r)-1-r^2)*(3+3*r^2-A(r)) = r*A(r)^2. At x>r, the equation (*) admits complex solutions for y. LINKS Table of n, a(n) for n=0..25. FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas. (1) (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0. (2) A(x) = 1 + x*A(x)^2 + x^2 + x^2*A(-x). (3) A(x) = 1 + x^2 + x*A(x)^2/A(-x). (4) A(x) = 1 + x^2/(1 - A(-x)). (5) A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x. (6) A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^2/G(-x) is the g.f. of A143339. Recurrence: (n-1)*(n+1)*(4*n^3 - 32*n^2 + 71*n - 30)*a(n) = 6*(8*n^3 - 56*n^2 + 101*n - 10)*a(n-1) + 6*(12*n^5 - 132*n^4 + 499*n^3 - 700*n^2 + 102*n + 305)*a(n-2) - 18*(n-4)*(8*n - 25)*a(n-3) + 27*(n-5)*(n-4)*(4*n^3 - 20*n^2 + 19*n + 13)*a(n-4). - Vaclav Kotesovec, Dec 29 2013 a(n) ~ c * 3^(n-1) * 2*sqrt(6*sqrt(3)-6 + sqrt(9+6*sqrt(3))) / (2*sqrt(Pi) * (2*sqrt(3)-3)^(n/2+1/4) * n^(3/2)), where c = 4/(2+12^(1/4)) if n is even and c = 12/(6+12^(3/4)) if n is odd. - Vaclav Kotesovec, Dec 29 2013 EXAMPLE G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +... A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +... where 1 - (1+x^2)/A(x) = x*A(x)/A(-x). Related expansions: A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +... A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +... where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x). PROG (PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^2/subst(A^2, x, -x)); polcoeff(A, n)} CROSSREFS Cf. A143339, A143554, A143556, A143557, A143558, A143559. Sequence in context: A059480 A105723 A280118 * A025234 A075308 A300461 Adjacent sequences: A143552 A143553 A143554 * A143556 A143557 A143558 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 24 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 15 22:50 EDT 2024. Contains 373412 sequences. (Running on oeis4.)