A110556 a(n) = binomial(2*n-1, n)*(-1)^n.

1, -1, 3, -10, 35, -126, 462, -1716, 6435, -24310, 92378, -352716, 1352078, -5200300, 20058300, -77558760, 300540195, -1166803110, 4537567650, -17672631900, 68923264410, -269128937220, 1052049481860, -4116715363800, 16123801841550

Offset: 0

Author

Reinhard Zumkeller, Jul 27 2005

Keywords

• sign
• easy

Examples

			1 - x + 3*x^2 - 10*x^3 + 35*x^4 - 126*x^5 + 462*x^6 - 1716*x^7 + 6435*x^8 - ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Another version of A001700, which is the main entry.

Cf. A000027, A000108, A000984, A007318, A033999, A088218, A110555.

Programs

Mathematica

Table[Binomial[2n-1,n](-1)^n,{n,0,30}] (* Harvey P. Dale, Apr 01 2012 *)
a[ n_] := (-1)^n Binomial[ 2 n - 1, n] (* Michael Somos, May 21 2013 *)
a[ n_] := Binomial[ -n, -2 n] (* Michael Somos, May 21 2013 *)
a[ n_] := SeriesCoefficient[(1 + x)^-n, {x, 0, n}] (* Michael Somos, May 21 2013 *)

PARI

{a(n) = if( n<0, 0, polcoeff( (1 + x)^-n + x * O(x^n), n))} /* Michael Somos, May 21 2013 */

Formula

a(n) = A110555(2*n,n), central terms in triangle A110555;

a(n) = A088218(n)*(-1)^n.

E.g.f.: E(x) = 1 - x/(G(0)+2*x) ; G(k) = (k+1)^2 - 2*x*(2*k+1) + 2*x*(2*k+3)*((k+1)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011

a(n) = coefficient of x^n of (1 / (1 + x))^n where 1 / (1 + x) is the g.f. of A033999. - Michael Somos, May 21 2013

HANKEL transform is A000027. HANKEL transform of a(n+1) is A033999(n+1). - Michael Somos, May 21 2013

E.g.f.: (1 + exp(-2*x) * BesselI(0,2*x)) / 2. - Ilya Gutkovskiy, Nov 03 2021

From Peter Bala, Feb 14 2024: (Start)

a(n) = binomial(-n, n).

O.g.f.: A(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) = 1/ (2 - c(-x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

The g.f. A(x) satisfies A(x/(1 - x)^2) = 1/(1 + x). (End)

a(n) ~ (-1)^n*2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 18 2024

D-finite with recurrence n*a(n) +2*(2*n-1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024