A014495 Central binomial coefficient - 1.

0, 0, 1, 2, 5, 9, 19, 34, 69, 125, 251, 461, 923, 1715, 3431, 6434, 12869, 24309, 48619, 92377, 184755, 352715, 705431, 1352077, 2704155, 5200299, 10400599, 20058299, 40116599, 77558759, 155117519, 300540194, 601080389, 1166803109, 2333606219, 4537567649

Offset: 0

Author

N. J. A. Sloane, Denis Pochuev (denis(AT)cdc.Informatik.TH-Darmstadt.de)

Keywords

• nonn
• easy

Comments

For n > 0: sum of positive elements in row (n-1) of triangle A214292. - Reinhard Zumkeller, Jul 12 2012

Number of Young tableaux with n cells and 2 rows. Also number of self-inverse permutations in S_n with longest increasing subsequence of length 2. The a(4) = 5 permutations are 1432, 2143, 3214, 3412, 4231 and the a(5) = 9 permutations are 15432, 21543, 32154, 35142, 42513, 43215, 45312, 52431, 53241. - Alois P. Heinz, Oct 03 2012

Number of nonempty subsets of {1,2,...,n} that contain the same number of even and odd numbers. For example, a(5)=9 and the 9 subsets are {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}. - Enrique Navarrete, Feb 10 2018

Links

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

Crossrefs

Cf. A001405, A037952 (first differences).

a(n) = A094718(n, n) = A094718(n-1, n)+1.

a(n) = A047884(n, 2) for n>=2. - Alois P. Heinz, Oct 03 2012

Cf. A214292.

Programs

Magma

[Binomial(n, Floor(n/2))-1: n in [0..50]]; // Vincenzo Librandi, Feb 11 2018

Maple

a:= n-> binomial(n, iquo(n, 2))-1:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 03 2012

Mathematica

Table[Binomial[n, Floor[n/2]] - 1, {n, 0, 50}] (* Bruno Berselli, Oct 03 2012 *)

Maxima

A014495(n):=binomial(n,floor(n/2))-1$
makelist(A014495(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

a(n) = A001405(n)-1.

a(n) = C(n,floor(n/2))-1. - Alois P. Heinz, Oct 03 2012

(n+1)*a(n)-2*a(n-1)-4*(n-1)*a(n-2) = 3*n-3 with n>1, a(0)=a(1)=0. - Bruno Berselli, Oct 03 2012

D-finite with recurrence: -(n+1)*(n-2)*a(n) +(n^2+n-4)*a(n-1) +2*(n-1)*(2*n-5)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0. - Conjectured by R. J. Mathar, Jan 04 2017, confirmed by Robert Israel, Feb 11 2018

G.f.: (x+1)/(2*x*(x-1)) - sqrt(1-4*x^2)/(2*x*(2*x-1)). - Robert Israel, Feb 11 2018

Extensions

Edited by Andrey Zabolotskiy, Apr 14 2025