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A037952
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a(n) = binomial(n, floor((n-1)/2)).
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28
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0, 1, 1, 3, 4, 10, 15, 35, 56, 126, 210, 462, 792, 1716, 3003, 6435, 11440, 24310, 43758, 92378, 167960, 352716, 646646, 1352078, 2496144, 5200300, 9657700, 20058300, 37442160, 77558760, 145422675, 300540195, 565722720, 1166803110, 2203961430, 4537567650
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OFFSET
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0,4
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COMMENTS
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First differences of central binomial coefficients: a(n) = A001405(n+1) - A001405(n).
The maximum size of an intersecting (or proper) antichain on an n-set. - Vladeta Jovovic, Dec 27 2000
Number of ordered trees with n+1 edges, having root of degree at least 2 and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch, Aug 02 2002
a(n)=number of Dyck (n+1)-paths that are symmetric but not prime. A prime Dyck path is one that returns to the x-axis only at its terminal point. For example a(3)=3 counts UDUUDDUD, UUDDUUDD, UDUDUDUD. - David Callan, Dec 09 2004
Number of involutions of [n+2] containing the pattern 132 exactly once. For example, a(3)=3 because we have 1'3'2'45, 42'5'13' and 52'4'3'1 (the entries corresponding to the pattern 132 are "primed"). - Emeric Deutsch, Nov 17 2005
Also number of ways to put n eggs in floor(n/2) baskets where order of the baskets matters and all baskets have at least 1 egg. - Ben Paul Thurston, Sep 30 2006
For n >= 1 the number of standard Young tableaux with shapes corresponding to partitions into at most 2 distinct parts. - Joerg Arndt, Oct 25 2012
It seems that 3, 4, 10, ... are Colbourn's Covering Array Numbers CAN(2,k,2). - Ryan Dougherty, May 27 2015
a(n) is the number of subsets of {1,2,...,n} that contain exactly 1 more odd than even elements. For example, for n = 6, a(6) = 15 and the 15 sets are {1}, {3}, {5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {3,4,5}, {3,5,6}, {1,2,3,4,5}, {1,2,3,5,6}, {1,3,4,5,6}. - Enrique Navarrete, Dec 21 2019
a(n) is the number of lattice paths of n steps taken from the step set {U=(1,1), D=(1,-1)} that start at the origin, never go below the x-axis, and end strictly above the x-axis; more succinctly, proper left factors of Dyck paths. For example, a(3)=3 counts UUU, UUD, UDU, and a(4)=4 counts UUUU, UUUD, UUDU, UDUU. - David Callan and Emeric Deutsch, Jan 25 2021
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LINKS
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FORMULA
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O.g.f.: (1-sqrt(1-4x^2))/(x - 2x^2 + x*sqrt(1-4x^2)).
D-finite with recurrence: (n+2)*a(n) + (-n-2)*a(n-1) + 2*(-2*n+1)*a(n-2) + 4*(n-2)*a(n-3) = 0. - R. J. Mathar, Jan 25 2013. Proved by Robert Israel, Nov 13 2014
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MAPLE
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a:= n-> binomial(n, floor((n-1)/2)):
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MATHEMATICA
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f[n_] := Binomial[n, Floor[(n-1)/2]]; Array[f, 35, 0] (* Robert G. Wilson v, Nov 13 2014 *)
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PROG
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(Haskell)
a037952 n = a037952_list !! n
a037952_list = zipWith (-) (tail a001405_list) a001405_list
(PARI) a(n) = binomial(n, (n-1)\2); \\ Altug Alkan, Oct 03 2018
(Magma) [Binomial(n, Floor((n-1)/2)): n in [0..40]]; // G. C. Greubel, Jun 21 2022
(SageMath) [binomial(n, (n-1)//2) for n in (0..40)] # G. C. Greubel, Jun 21 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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