A128015 Binomial coefficients C(2n+1,n) repeated.

1, 1, 3, 3, 10, 10, 35, 35, 126, 126, 462, 462, 1716, 1716, 6435, 6435, 24310, 24310, 92378, 92378, 352716, 352716, 1352078, 1352078, 5200300, 5200300, 20058300, 20058300, 77558760, 77558760, 300540195, 300540195

Offset: 0

Author

Paul Barry, Feb 11 2007

Keywords

• easy
• nonn

Comments

Hankel transform is A128017. Binomial transform is A005717(n+1).

Crossrefs

Cf. A000108, A001700, A005717, A128017.

Programs

Mathematica

With[{c=Table[Binomial[2n+1,n],{n,0,20}]},Riffle[c,c]] (* Harvey P. Dale, May 02 2012 *)

Formula

G.f.: (1+x)*c(x^2)/sqrt(1-4x^2), c(x) the g.f. of A000108.

E.g.f.: exp(-x)*dif(exp(x)*Bessel_I(1,2x),x).

a(n) = C(n+1, n/2)*(1+(-1)^n)/2 + C(n, (n-1)/2)*(1-(-1)^n)/2; as moment sequence a(n) = (1/(2*Pi))*Integral_{x=-2..2} x^n*x*(1+x)/sqrt(4-x^2).

D-finite with recurrence: -(n+2)*(3*n-1)*a(n) - 4*a(n-1) + 4*n*(3*n+2)*a(n-2) = 0. - R. J. Mathar, Jun 17 2016

a(n) = A001700(floor(n/2)). - Georg Fischer, Nov 28 2022