%I A001405 M0769 N0294
%S A001405 1,1,2,3,6,10,20,35,70,126,252,462,924,1716,3432,6435,12870,24310,
%T A001405 48620,92378,184756,352716,705432,1352078,2704156,5200300,10400600,
%U A001405 20058300,40116600,77558760,155117520,300540195,601080390,1166803110
%N A001405 Central binomial coefficients: C(n,floor(n/2)).
%C A001405 By symmetry, a(n)=C(n,ceiling(n/2)). - Labos E. (labos(AT)ana.sote.hu), 
               Mar 20 2003
%C A001405 Sperner's theorem says that this is the maximal number of subsets of 
               an n-set such that no one contains another.
%C A001405 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
%C A001405 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
%C A001405 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
%C A001405 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
%C A001405 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
%C A001405 a(n) is odd iff n=2^k-1 - Jon Perry (perry(AT)globalnet.co.uk), May 05 
               2005
%C A001405 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
%C A001405 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
%C A001405 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
%C A001405 Also the number of standard tableaux of height less than or equal to 
               2. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Mar 24 2007
%C A001405 Hankel transform of this sequence forms A000012 = [1,1,1,1,1,1,1,...] 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A001405 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
%C A001405 Equals right border of triangle A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 28 2008]
%D A001405 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.
%D A001405 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 
               2544-2563.
%D A001405 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 
               1999; see p. 135.
%D A001405 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.
%D A001405 K. Engel, Sperner Theory, Camb. Univ. Press, 1997; Theorem 1.1.1.
%D A001405 P. Frankl, Extremal sets systems, Chap. 24 of R. L. Graham et al., eds, 
               Handbook of Combinatorics, North-Holland.
%D A001405 D. Lubell, A short proof of Sperner's lemma, J. Combin. Theory, 1 (1966), 
               299.
%D A001405 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society 
               Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A001405 M. A. Narcowich, Problem 73-6, SIAM Review, Vol. 16, No. 1, 1974, p. 
               97.
%D A001405 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001405 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001405 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 7.16(b), p. 452.
%D A001405 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.
%H A001405 T. D. Noe, <a href="b001405.txt">Table of n, a(n) for n = 0..200</a>
%H A001405 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001405 H. Bottomley, <a href="a001405.gif">Illustration of initial terms</a>
%H A001405 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               77
%H A001405 J. R. Griggs, <a href="http://arXiv.org/abs/math.NT/9304211">On the distribution 
               of sums of residues</a>
%H A001405 O. Guibert and T. Mansour, <a href="http://www.mat.univie.ac.at/~slc/
               opapers/s48guimans.html">Restricted 132-involutions</a>
%H A001405 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</
               a>
%H A001405 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A001405 P. Leroux and E. Rassart, <a href="http://arXiv.org/abs/math.CO/9901135">
               [math/9901135] Enumeration of Symmetry Classes of Parallelogram Polyominoes</
               a>
%H A001405 D. Merlini, <a href="http://www.dmtcs.org/pspapers/dmAC0121.ps">Generating 
               functions for the area below some lattice paths</a>
%H A001405 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A001405 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CentralBinomialCoefficient.html">Link to a section of The World of 
               Mathematics.</a>
%H A001405 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               QuotaSystem.html">Link to a section of The World of Mathematics.</
               a>
%H A001405 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001405 a(n) = Max C(n, k), 1 <= k <= n.
%F A001405 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.
%F A001405 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
%F A001405 G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 28 2003
%F A001405 E.g.f.: BesselI(0, 2*x)+BesselI(1, 2*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 28 2003
%F A001405 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
%F A001405 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
%F A001405 a(n)=sum{k=0..floor(n/2), binomial(n, k)*binomial(1, n-2k)}. - Paul Barry 
               (pbarry(AT)wit.ie), May 13 2005
%F A001405 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
%F A001405 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
%F A001405 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
%F A001405 a(n)=Sum_{k, 0<=k<=n}A120730(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 16 2008]
%F A001405 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
%F A001405 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]
%p A001405 A001405 := n->binomial(n,floor(n/2));
%t A001405 Table[Binomial[n, Floor[n/2]], {n, 0, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 08 2006
%o A001405 (PARI) a(n)=binomial(n,n\2)
%Y A001405 Cf. A051920. a(2*n)= A000984(n), a(2*n+1)= A001700(n). Row sums of Catalan 
               triangle A053121.
%Y A001405 Enumerates the structures encoded by A061854 and A061855.
%Y A001405 First differences are in A037952.
%Y A001405 Apparently a(n) = lim[k=1..inf, A094718(k, n)].
%Y A001405 Cf. A001006, A005817, A049401, A007579, A007578.
%Y A001405 Cf. A005773, A001700, A132815.
%Y A001405 Partial sums are in A036256.
%Y A001405 A153585 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2008]
%Y A001405 Sequence in context: A026034 A037031 A056202 this_sequence A126930 A036557 
               A047131
%Y A001405 Adjacent sequences: A001402 A001403 A001404 this_sequence A001406 A001407 
               A001408
%K A001405 nonn,easy,nice,core
%O A001405 0,3
%A A001405 N. J. A. Sloane (njas(AT)research.att.com).