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 A014495 Central binomial coefficient - 1. 9

%I

%S 0,0,1,2,5,9,19,34,69,125,251,461,923,1715,3431,6434,12869,24309,

%T 48619,92377,184755,352715,705431,1352077,2704155,5200299,10400599,

%U 20058299,40116599,77558759,155117519,300540194,601080389,1166803109,2333606219,4537567649

%N Central binomial coefficient - 1.

%C For n > 0: sum of positive elements in row (n-1) of triangle A214292. - _Reinhard Zumkeller_, Jul 12 2012

%C 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

%C 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

%H Alois P. Heinz, <a href="/A014495/b014495.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = C(n-1,floor((n-1)/2))-1. - _Alois P. Heinz_, Oct 03 2012

%F n*a(n)-2*a(n-1)-4*(n-2)*a(n-2) = 3*n-6 with n>1, a(0)=a(1)=0. - _Bruno Berselli_, Oct 03 2012

%F 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

%F G.f.: (x+1)/(2*x*(x-1)) - sqrt(1-4*x^2)/(2*x*(2*x-1)). - _Robert Israel_, Feb 11 2018

%p a:= n-> binomial(n, iquo(n, 2))-1:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 03 2012

%t Table[Binomial[n - 1, Floor[(n - 1)/2]] - 1, {n, 0, 50}] (* _Bruno Berselli_, Oct 03 2012 *)

%o (Maxima) A014495(n):=binomial(n-1,floor((n-1)/2))-1\$

%o makelist(A014495(n),n,1,30); /* _Martin Ettl_, Nov 01 2012 */

%o (MAGMA) [Binomial(n-1, Floor((n-1)/2))-1: n in [1..50]]; // _Vincenzo Librandi_, Feb 11 2018

%Y Cf. A001405.

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

%Y a(n) = A047884(n, 2) for n>=2. - _Alois P. Heinz_, Oct 03 2012

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, denis(AT)cdc.Informatik.TH-Darmstadt.DE (Denis Pochuev)

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Last modified February 20 17:04 EST 2020. Contains 332080 sequences. (Running on oeis4.)