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

0,3

Comments

Old name was: Central coefficients of number triangle A119326.

Links

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

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

Example

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

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 *)

Crossrefs

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

Column k=2 of A318557.

Sequence in context: A177173 A046241 A048499 * A110148 A281199 A081956

Adjacent sequences: A119355 A119356 A119357 * A119359 A119360 A119361

Keyword

easy,nonn

Author

Paul Barry, May 16 2006

Extensions

New name from Alois P. Heinz, Feb 04 2017

Status

approved