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%I #132 Jul 06 2023 01:55:58
%S 0,1,1,3,4,10,15,35,56,126,210,462,792,1716,3003,6435,11440,24310,
%T 43758,92378,167960,352716,646646,1352078,2496144,5200300,9657700,
%U 20058300,37442160,77558760,145422675,300540195,565722720,1166803110,2203961430,4537567650
%N a(n) = binomial(n, floor((n-1)/2)).
%C First differences of central binomial coefficients: a(n) = A001405(n+1) - A001405(n).
%C The maximum size of an intersecting (or proper) antichain on an n-set. - _Vladeta Jovovic_, Dec 27 2000
%C 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
%C 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
%C 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
%C 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
%C 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
%C It seems that 3, 4, 10, ... are Colbourn's Covering Array Numbers CAN(2,k,2). - _Ryan Dougherty_, May 27 2015
%C For n > 0: a(n) = A265848(n,0). - _Reinhard Zumkeller_, Dec 24 2015
%C Essentially the same as A007007. - _Georg Fischer_, Oct 02 2018
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A037952/b037952.txt">Table of n, a(n) for n = 0..1000</a>
%H Cyril Banderier and Michael Wallner, <a href="https://arxiv.org/abs/1707.01931">Lattice paths with catastrophes</a>, arXiv:1707.01931 [math.CO], 2017.
%H J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%H Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p398.pdf">Dyck Paths with catastrophes modulo the positions of a given pattern</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
%H Jean-Luc Baril and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/pathwall.pdf">Fibonacci and Catalan paths in a wall</a>, 2023.
%H C. J. Colbourn, <a href="http://www.public.asu.edu/~ccolbou/src/tabby/2-2-ca.html">Table of CAN(2, k, 2)</a>
%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.021">Ordered trees with prescribed root degrees, node degrees and branch lengths</a>, Discrete Math., 282, 2004, 89-94.
%H O. Guibert and T. Mansour, <a href="http://www.emis.de/journals/SLC/wpapers/s48guimans.html">Restricted 132-involutions</a>, Sem. Lotharingien de Combinatoire, 48, 2002, Article B48a (Corollary 4.2).
%H M. Miyakawa, A. Nozaki, G. Pogosyan, and I. G. Rosenberg, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00054-2">A map from the lower-half of the n-Cube onto the (n-1)-Cube which preserves intersecting antichains</a>, Discr. Appl. Math. 92 (2-3) (1999) 223-228.
%H M. van de Vel, <a href="http://www.emis.de/journals/JACO/Volume9_2/g618g3480371x5m8.html">Determination of msd(L^n)</a>, J. Algebraic Combin., 9 (1999), 161-171.
%F E.g.f.: BesselI(1, 2*x) + BesselI(2, 2*x). - _Vladeta Jovovic_, Apr 28 2003
%F O.g.f.: (1-sqrt(1-4x^2))/(x - 2x^2 + x*sqrt(1-4x^2)).
%F Convolution of A001405 and A126120 shifted right: g001405(x)*g126120(x) = g037952(x)/x. - _Philippe Deléham_, Mar 17 2007
%F 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
%F a(n) = binomial(n, (n-2)/2) = A001791(n/2), n even; a(n) = binomial(n, (n+1)/2) = A001700((n-1)/2), n odd. - _Enrique Navarrete_, Dec 21 2019
%F A001405(n) = a(n) + A000108(n/2), where A(.)=0 for non-integer arguments. - _R. J. Mathar_, Sep 23 2021
%F a(n) = Sum_{m=1..n} A053121(n,m) [comment Callan-Deutsch]. - _R. J. Mathar_, Sep 23 2021
%F a(2n+1) = A000984(n+1)/2. - _R. J. Mathar_, Sep 23 2021
%F a(n) = Sum_{k=2..n} A143359(n,k). [Callan's 2004 comment]. - _R. J. Mathar_, Sep 24 2021
%p a:= n-> binomial(n, floor((n-1)/2)):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Sep 19 2017
%t Table[ Binomial[n, Floor[n/2]], {n, 0, 35}]//Differences (* _Jean-François Alcover_, Jun 10 2013 *)
%t f[n_] := Binomial[n, Floor[(n-1)/2]]; Array[f, 35, 0] (* _Robert G. Wilson v_, Nov 13 2014 *)
%o (Haskell)
%o a037952 n = a037952_list !! n
%o a037952_list = zipWith (-) (tail a001405_list) a001405_list
%o -- _Reinhard Zumkeller_, Mar 04 2012
%o (PARI) a(n) = binomial(n, (n-1)\2); \\ _Altug Alkan_, Oct 03 2018
%o (Magma) [Binomial(n, Floor((n-1)/2)): n in [0..40]]; // _G. C. Greubel_, Jun 21 2022
%o (SageMath) [binomial(n, (n-1)//2) for n in (0..40)] # _G. C. Greubel_, Jun 21 2022
%Y Cf. A007007, A032263, A014495 (partial sums), A001405 (partial sums + 1).
%Y Cf. A035951, A035953, A035954, A035955, A035956, A035957.
%Y Cf. A051303, A051304, A051305, A051306, A051307.
%Y Cf. A047171, A036256, A051920.
%Y Cf. A265848.
%K nonn
%O 0,4
%A _N. J. A. Sloane_
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