A128057 Expansion of (1+x)/sqrt(1+4x^2).

1, 1, -2, -2, 6, 6, -20, -20, 70, 70, -252, -252, 924, 924, -3432, -3432, 12870, 12870, -48620, -48620, 184756, 184756, -705432, -705432, 2704156, 2704156, -10400600, -10400600, 40116600, 40116600, -155117520, -155117520

Offset: 0

Author

Paul Barry, Feb 13 2007

Keywords

• easy
• sign

Comments

Hankel transform is A128056. Binomial transform is A128058. Unsigned version is A128014.

Crossrefs

Cf. A000984.

Programs

Mathematica

CoefficientList[Series[(1+x)/Sqrt[1+4x^2],{x,0,40}],x] (* Harvey P. Dale, May 12 2015 *)

Formula

a(n)=(-1)^C(n,2)*(C(n,n/2)*(1+(-1)^n)/2+C(n-1,(n-1)/2)*(1-(-1)^n)/2);

Conjecture: n*(5*n-9)*a(n) +4*a(n-1) +4*(5*n-4)*(n-2)*a(n-2)=0. - R. J. Mathar, Dec 02 2014

D-finite with recurrence: n*a(n) +(n-2)*a(n-1) +4*(n-1)*a(n-2) +4*(n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2014