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%I #22 Sep 08 2022 08:44:56
%S 0,0,0,2,3,9,14,34,55,125,209,461,791,1715,3002,6434,11439,24309,
%T 43757,92377,167959,352715,646645,1352077,2496143,5200299,9657699,
%U 20058299,37442159,77558759,145422674,300540194,565722719,1166803109,2203961429,4537567649
%N Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= (n-1)/2.
%C For n>=1 the number of standard Young tableaux with shapes corresponding to partitions into two distinct parts. - _Joerg Arndt_, Oct 25 2012
%H Alois P. Heinz, <a href="/A047171/b047171.txt">Table of n, a(n) for n = 0..1000</a>
%F 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.
%p a:= n-> binomial(n, iquo(n-1,2))-1:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Nov 17 2012
%t a[n_] := Binomial[n, Floor[(n-1)/2]]-1; a[0] = 0; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 03 2015 *)
%o (Magma) [0] cat [Binomial(n, Floor((n-1)/2))-1: n in [1..40]]: // _Vincenzo Librandi_, Jul 03 2015
%Y Column k=2 of A219311. - _Alois P. Heinz_, Nov 17 2012
%K nonn
%O 0,4
%A _Clark Kimberling_
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