A119358 Number of n-element subsets of [2n] having an even sum.

1, 1, 2, 10, 38, 126, 452, 1716, 6470, 24310, 92252, 352716, 1352540, 5200300, 20056584, 77558760, 300546630, 1166803110, 4537543340, 17672631900, 68923356788, 269128937220, 1052049129144, 4116715363800, 16123803193628, 63205303218876, 247959261273752

Offset: 0

Author

Paul Barry, May 16 2006

Keywords

• easy
• nonn

Comments

Old name was: Central coefficients of number triangle A119326.

Examples

			a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - _Alois P. Heinz_, Feb 04 2017
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A071688, A119326, A119359, A169888, A282011.

Column k=2 of A318557.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
     ((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
      /(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 26 2018

Mathematica

Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

Formula

G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.

a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.

a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.

a(n) = A119326(2n,n).

a(n) = A071688(n) + A119359(n) for n>=1.

D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 05 2012

a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 04 2016

a(n) = A282011(2n,n). - Alois P. Heinz, Feb 04 2017

Extensions

New name from Alois P. Heinz, Feb 04 2017