A037952 - OEIS

A037952

a(n) = binomial(n, floor((n-1)/2)).

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

0,4

COMMENTS

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

Essentially the same as A007007. - Georg Fischer, Oct 02 2018

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

For n >= 3, a(n) is also the number of pinnacle sets in the (n-2)-Plummer-Toft graph. - Eric W. Weisstein, Sep 11 2024

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.

Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.

C. J. Colbourn, Table of CAN(2, k, 2)

Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 89-94.

O. Guibert and T. Mansour, Restricted 132-involutions, Sem. Lotharingien de Combinatoire, 48, 2002, Article B48a (Corollary 4.2).

M. Miyakawa, A. Nozaki, G. Pogosyan, and I. G. Rosenberg, A map from the lower-half of the n-Cube onto the (n-1)-Cube which preserves intersecting antichains, Discr. Appl. Math. 92 (2-3) (1999) 223-228.

M. van de Vel, Determination of msd(L^n), J. Algebraic Combin., 9 (1999), 161-171.

Eric Weisstein's World of Mathematics, Pinnacle Set.

Eric Weisstein's World of Mathematics, Plummer-Toft Graph.

FORMULA

E.g.f.: BesselI(1, 2*x) + BesselI(2, 2*x). - Vladeta Jovovic, Apr 28 2003

O.g.f.: (1-sqrt(1-4x^2))/(x - 2x^2 + x*sqrt(1-4x^2)).

Convolution of A001405 and A126120 shifted right: g001405(x)*g126120(x) = g037952(x)/x. - Philippe Deléham, Mar 17 2007

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

For n > 0: a(n) = A265848(n,0). - Reinhard Zumkeller, Dec 24 2015

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

From R. J. Mathar, Sep 23 2021: (Start)

A001405(n) = a(n) + A000108(n/2), where A(.)=0 for non-integer arguments.

a(n) = Sum_{m=1..n} A053121(n,m) [comment Callan-Deutsch].

a(2n+1) = A000984(n+1)/2. (End)

a(n) = Sum_{k=2..n} A143359(n,k). [Callan's 2004 comment]. - R. J. Mathar, Sep 24 2021

From Amiram Eldar, Sep 27 2024: (Start)

Sum_{n>=1} 1/a(n) = 1 + Pi/sqrt(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = (3 - Pi/sqrt(3))/9. (End)

MAPLE

a:= n-> binomial(n, floor((n-1)/2)):