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A037952 a(n) = binomial(n, floor((n-1)/2)). 21
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 (list; graph; refs; listen; history; text; internal format)
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

[1,1,3,4,10,15,35,56,...] is the convolution of A001405 with A126120. - Philippe Deléham, Mar 17 2007

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

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

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

LINKS

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

Cyril Banderier, 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.

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, 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.

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)).

Conjecture: (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

Conjecture proved by Robert Israel, Nov 13 2014

MAPLE

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

seq(a(n), n=0..35);  # Alois P. Heinz, Sep 19 2017

MATHEMATICA

Table[ Binomial[n, Floor[n/2] ], {n, 0, 35}] // Differences (* Jean-François Alcover, Jun 10 2013 *)

f[n_] := Binomial[n, Floor[(n - 1)/2]]; Array[f, 35, 0] (* Robert G. Wilson v, Nov 13 2014 *)

PROG

(Haskell)

a037952 n = a037952_list !! n

a037952_list = zipWith (-) (tail a001405_list) a001405_list

-- Reinhard Zumkeller, Mar 04 2012

(PARI) a(n) = binomial(n, (n-1)\2); \\ Altug Alkan, Oct 03 2018

CROSSREFS

Cf. A007007, A032263, A051303-A051307, A001405.

Cf. A047171, A036256, A051920.

Cf. A265848.

Sequence in context: A054184 A188022 A007007 * A281903 A093512 A081160

Adjacent sequences:  A037949 A037950 A037951 * A037953 A037954 A037955

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 18 07:00 EDT 2018. Contains 316307 sequences. (Running on oeis4.)