A198786 G.f. satisfies: A(x) = 1 + 2*x*sqrt(A(x)/A(-x)).

1, 2, 4, 4, 0, -4, 0, 8, 0, -20, 0, 56, 0, -168, 0, 528, 0, -1716, 0, 5720, 0, -19448, 0, 67184, 0, -235144, 0, 832048, 0, -2971600, 0, 10697760, 0, -38779380, 0, 141430680, 0, -518579160, 0, 1910554800, 0, -7069052760, 0, 26256481680, 0, -97865068080, 0, 365930254560, 0

Offset

0,2

Links

Table of n, a(n) for n=0..48.

Formula

G.f. A(x) satisfies:

(1) A(x) * A(-x) = 1 + 4*x^2.

(2) A(x) / A(-x) = 2*A(x) - 1.

(3) A(x) + A(-x) = 2 + 8*x^2.

(4) A(x) = 1 + 2*x + 4*x^2 + 4*x^3*C(-x^2), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

a(n) = 4*hypergeom([3-n, 2-n], [2], -1)) for n>0. - Peter Luschny, May 22 2017

D-recurrence: -(n+2)*(n-1)*a(n) -(n+3)*(n-2)*a(n-1) -4*(n+2)*(n-4)*a(n-2) -4*(n+3)*(n-5)*a(n-3)=0. - R. J. Mathar, May 22 2019

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 4*x^3 - 4*x^5 + 8*x^7 - 20*x^9 + 56*x^11 +...
where
A(x)/A(-x) = 1 + 4*x + 8*x^2 + 8*x^3 - 8*x^5 + 16*x^7 - 40*x^9 + 112*x^11 +...

Maple

a := n -> `if`(n=0, 1, 4*hypergeom([3-n, 2-n], [2], -1)):
seq(simplify(a(n)), n=0..48); # Peter Luschny, May 22 2017

Mathematica

m = 50; A[_] = 1; Do[A[x_] = 1 + 2 x Sqrt[A[x]/A[-x]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jun 29 2019 *)

Prog

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

Crossrefs

Sequence in context: A111172 A173556 A247554 * A176531 A198362 A197827

Adjacent sequences: A198783 A198784 A198785 * A198787 A198788 A198789

Keyword

sign

Author

Paul D. Hanna, Oct 29 2011

Status

approved