A047171 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= (n-1)/2.

0, 0, 0, 2, 3, 9, 14, 34, 55, 125, 209, 461, 791, 1715, 3002, 6434, 11439, 24309, 43757, 92377, 167959, 352715, 646645, 1352077, 2496143, 5200299, 9657699, 20058299, 37442159, 77558759, 145422674, 300540194, 565722719, 1166803109, 2203961429, 4537567649

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

For n>=1 the number of standard Young tableaux with shapes corresponding to partitions into two distinct parts. - Joerg Arndt, Oct 25 2012

Links

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

Crossrefs

Column k=2 of A219311. - Alois P. Heinz, Nov 17 2012

Programs

Magma

[0] cat [Binomial(n, Floor((n-1)/2))-1: n in [1..40]]; // Vincenzo Librandi, Jul 03 2015

Maple

a:= n-> binomial(n, iquo(n-1,2))-1:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 17 2012

Mathematica

a[n_] := Binomial[n, Floor[(n-1)/2]]-1; a[0] = 0; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 03 2015 *)

Formula

a(n) = A037952(n) - 1. Proof by Ira Gessel: Write down the number of such subsets with k elements <= (n-1)/2 as a product of two binomial coefficients, then evaluate the sum using Vandermonde's theorem.