%I A005700 M2975
%S A005700 1,1,3,14,84,594,4719,40898,379236,3711916,37975756,403127256,
%T A005700 4415203280,49671036900,571947380775,6721316278650,80419959684900,
%U A005700 977737404590100,12058761323277900,150656212896017400
%N A005700 Number of Dyck paths: a(n) = number of walks of 2n unit steps north, 
               east, south, or west starting and ending at the origin and confined 
               to the first octant.
%C A005700 Image of Catalan numbers (A000108) under "little Hankel" transform that 
               sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.
%C A005700 The Niederhausen reference counts various classes of first octant paths 
               by number of contacts with the line y=x. - David Callan (callan(AT)stat.wisc.edu), 
               Sep 18 2007
%D A005700 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005700 N. Bonichon, A bijection between realizers of maximal plane graphs and 
               pairs of non-crossing Dyck paths, Discr. Math., 298 (2005), 104-114.
%D A005700 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, 
               Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see 
               p. 183).
%D A005700 D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 
               112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes 
               Math. 1234, 1986.
%D A005700 Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in 
               the First Octant by Their Number of Contacts with the Diagonal, Journal 
               of Integer Sequences, Vol. 8 (2005), Article 05.4.3.
%H A005700 W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, 
               <a href="http://arXiv.org/abs/math.CO/0501230">Crossings and nestings 
               of matchings and partitions</a>
%H A005700 Alec Mihailovs, <a href="http://mihailovs.com/Alec/papers.html">Enumeration 
               of walks on lattices</a>.
%H A005700 <a href="Sindx_Y.html#Young">Index entries for sequences related to Young 
               tableaux.</a>
%H A005700 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>.
%F A005700 G.f.: 3_F_2( [ 1, 1/2, 3/2 ]; [ 3, 4 ]; 16 x ).
%F A005700 a(n) = 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!) (Mihailovs).
%F A005700 a(n)=Det[Table[binomial[i+1, j-i+2], {i, 1, n}, {j, 1, n}]] - David Callan 
               (callan(AT)stat.wisc.edu), Jul 20 2005
%F A005700 a(n)=b(n)b(n+1)/6 where b(n) is the superballot number A007054. - David 
               Callan (callan(AT)stat.wisc.edu), Feb 01 2007
%F A005700 a(n)=A000108(n)*A000108(n+2)-A000108(n+1)^2. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 11 2007
%F A005700 G.f.: (1/(4*x^2)) * (1+6*x - (16*x-1)^4*hypergeom([5/2, 7/2],[2],16*x)) 
               [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 02 2009]
%e A005700 Example: a(2)=3 counts EWEW, EEWW, ENSW.
%Y A005700 See 138349 for another version.
%Y A005700 Sequence in context: A121687 A154757 A074535 this_sequence A088717 A111538 
               A088716
%Y A005700 Adjacent sequences: A005697 A005698 A005699 this_sequence A005701 A005702 
               A005703
%K A005700 nonn,easy,new
%O A005700 0,3
%A A005700 N. J. A. Sloane (njas(AT)research.att.com).
%E A005700 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999
%E A005700 Corrected by Vladeta Jovovic (vladeta(AT)eunet.rs), May 23 2004
%E A005700 Better definition from David Callan (callan(AT)stat.wisc.edu), Sep 18 
               2007