A110145 a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).

0, 1, 4, 10, 32, 126, 472, 1716, 6400, 24310, 92504, 352716, 1351616, 5200300, 20060016, 77558760, 300533760, 1166803110, 4537591960, 17672631900, 68923172032, 269128937220, 1052049834576, 4116715363800, 16123800489472, 63205303218876, 247959271674352

Offset: 0

Author

Paul Barry, Jul 13 2005

Keywords

• easy
• nonn

Comments

Interleaves A002458 and A037964.

Number of n-element subsets of [2n] having an odd sum. - Alois P. Heinz, Feb 06 2017

Links

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

Crossrefs

Cf. A159916.

Programs

Mathematica

Table[Sum[Binomial[n,k]^2 Mod[k,2],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Feb 21 2013 *)
Table[(Binomial[2 n, n] - Binomial[n, n/2] Cos[Pi n/2])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 04 2016 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)^2*(k % 2)); \\ Michel Marcus, Oct 05 2016

Formula

a(n) = Sum_{k=0..n} C(n, k)^2*(1-(-1)^k)/2.

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

a(n) = (binomial(2*n, n) - binomial(n, n/2)*cos(Pi*n/2))/2 = n^2 * hypergeom([1/2-n/2, 1/2-n/2, 1-n/2, 1-n/2], [1, 3/2, 3/2], 1). - Vladimir Reshetnikov, Oct 04 2016

a(n) = A159916(2n,n). - Alois P. Heinz, Feb 06 2017