0,3

By symmetry, a(n)=C(n,ceiling(n/2)). - Labos E. (labos(AT)ana.sote.hu), Mar 20 2003

Sperner's theorem says that this is the maximal number of subsets of an n-set such that no one contains another.

When computed from index -1, [seq(binomial(n,floor(n/2)), n=-1..30)]; -> [1,1,1,2,3,6,10,20,35,70,126,...] and convolved with aerated Catalans [seq((n+1 mod 2)*binomial(n,n/2)/((n/2)+1), n=0..30)]; -> [1,0,1,0,2,0,5,0,14,0,42,0,132,0,...] shifts left by one: [1,1,2,3,6,10,20,35,70,126,252,...] and if again convolved with aerated Catalans, seems to give A037952 apart from the initial term. - Antti Karttunen, Jun 05 2001

Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

Gives for n>=1 the maximum absolute column sum norm of the inverse of the Vandermonde matrix (a_ij) i=0..n-1, j=0..n-1 with a_00=1 and a_ij=i^j for (i,j)!=(0,0). - Torsten Muetze (torstenmuetze(AT)gmx.de), Feb 06 2004

Image of Catalan numbers A000108 under the Riordan array (1/(1-2x),-x/(1-2x)) or A065109. - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005

Number of left factors of Dyck paths, consisting of n steps. Example: a(4)=6 because we have UDUD, UDUU, UUDD, UUDU, UUUD and UUUU, where U=(1,1) and D=(1,-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 23 2005

a(n) is odd iff n=2^k-1 - Jon Perry (perry(AT)globalnet.co.uk), May 05 2005

An inverse Chebyshev transform of binomial(1,n)=(1,1,0,0,0,...) where g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), with c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 13 2005

In a random walk on the number line, starting at 0 and with 0 absorbing after the first step, number of ways of ending up at a positive integer after n steps. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jul 31 2005

Maximum number of sums of the form sum(0<i<=n, (e(i)*a(i))) that are congruent to 0 mod q, where e_i=0 or 1 and GCD(a_i,q)=1, provided that q>ceil(n/2). - Ralf Stephan, Apr 27 2003

Also the number of standard tableaux of height less than or equal to 2. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Mar 24 2007

Hankel transform of this sequence forms A000012 = [1,1,1,1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007

A001263 * [1, -2, 3, -4, 5,...] = (1, -1, -2, 3, 6, -10, -20, 35, 70, -126,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 02 2008

Equals right border of triangle A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 135.

F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

K. Engel, Sperner Theory, Camb. Univ. Press, 1997; Theorem 1.1.1.

P. Frankl, Extremal sets systems, Chap. 24 of R. L. Graham et al., eds, Handbook of Combinatorics, North-Holland.

D. Lubell, A short proof of Sperner's lemma, J. Combin. Theory, 1 (1966), 299.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

M. A. Narcowich, Problem 73-6, SIAM Review, Vol. 16, No. 1, 1974, p. 97.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), p. 452.

P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n = 0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

H. Bottomley, Illustration of initial terms

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 77

J. R. Griggs, On the distribution of sums of residues

O. Guibert and T. Mansour, Restricted 132-involutions

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

P. Leroux and E. Rassart, [math/9901135] Enumeration of Symmetry Classes of Parallelogram Polyominoes

D. Merlini, Generating functions for the area below some lattice paths

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

a(n) = Max C(n, k), 1 <= k <= n.

G.f.: (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)); where c(x) = g.f. for Catalan numbers A000108.

a(0) = 1; a(2m+2) = 2a(2m+1); a(2m+1) = sum((-1)^k*a(k)*a(2m-k), k = 0..2m). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001

G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 28 2003

E.g.f.: BesselI(0, 2*x)+BesselI(1, 2*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 28 2003

a(0) = 1; a(2m+2) = 2a(2m+1); a(2m+1) = 2a(2m) - c(m), where c(m)=A000108(m) are the Catalan numbers. - Christopher Hanusa (chanusa(AT)washington.edu), Nov 25 2003

a(n)=sum{k=0..n, (-1)^k*2^(n-k)*binomial(n, k)*A000108(k)} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005

a(n)=sum{k=0..floor(n/2), binomial(n, k)*binomial(1, n-2k)}. - Paul Barry (pbarry(AT)wit.ie), May 13 2005

a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, k)(cos((n-2k+1)*pi/2)+sin((n-2k+1)*pi/2))}; a(n)=sum{k=0..n+1, binomial(n+1, (n-k+1)/2)(1-(-1)^(n-k))(cos(k*pi/2)+sin(k*pi))/2}. - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004

a(n)=sum{k=floor(n/2)..n, C(n,n-k)-C(n,n-k-1)}. - Paul Barry (pbarry(AT)wit.ie), Sep 06 2007

Inverse binomial transform of A005773 starting (1, 2, 5, 13, 35, 96,...) and double inverse binomial transform of A001700. Row sums of triangle A132815. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 31 2007

a(n)=Sum_{k, 0<=k<=n}A120730(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 16 2008]

a(n)=sum{k=0 to floor(n/2), C(n,n-k)-C(n,n-k-1)}. - Nishant Doshi (doshinikki2004(AT)gmail.com), Apr 06 2009

G.f.: 1/(1-x-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Aug 12 2009]

A001405 := n->binomial(n, floor(n/2));

Table[Binomial[n, Floor[n/2]], {n, 0, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006

(PARI) a(n)=binomial(n, n\2)

Cf. A051920. a(2*n)= A000984(n), a(2*n+1)= A001700(n). Row sums of Catalan triangle A053121.

Enumerates the structures encoded by A061854 and A061855.

First differences are in A037952.

Apparently a(n) = lim[k=1..inf, A094718(k, n)].

Cf. A001006, A005817, A049401, A007579, A007578.

Cf. A005773, A001700, A132815.

Partial sums are in A036256.

A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]

Sequence in context: A026034 A037031 A056202 this_sequence A126930 A036557 A047131

Adjacent sequences: A001402 A001403 A001404 this_sequence A001406 A001407 A001408

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).