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 A005700 a(n) = C(n)*C(n+2) - C(n+1)^2 where C() are the Catalan numbers A000108. (Formerly M2975) 24
 1, 1, 3, 14, 84, 594, 4719, 40898, 379236, 3711916, 37975756, 403127256, 4415203280, 49671036900, 571947380775, 6721316278650, 80419959684900, 977737404590100, 12058761323277900, 150656212896017400, 1904342169333848400, 24328661192286773400, 313839729380499376860 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The old name was: Number of closed walks of 2n unit steps north, east, south, or west starting and ending at the origin and confined to the first octant. 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}. The Niederhausen reference counts various classes of first octant paths by number of contacts with the line y=x. - David Callan, Sep 18 2007 In Sloane and Plouffe (1995) this was incorrectly described as "Dyck paths". Also matchings avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013 From Bruce Westbury, Aug 22 2013: (Start) a(n) is also the number of nested pairs of Dyck paths of length n starting and ending at the origin; a(n) is also the number of 3-noncrossing perfect matchings on 2n points; a(n) is also the number of 2-triangulations on n-gon; a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the spin representation of Spin(5); a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the defining representation of Sp(4). (End) a(-1) = -3/2, a(-2) = -1/4 makes some formulas true for all n in Z. - Michael Somos, Oct 02 2014 a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the pattern 312. - Colin Defant, Jun 08 2019 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012. See Table 1. N. Bonichon, A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths, Discr. Math., 298 (2005), 104-114. Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020. See Section 2. W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005. S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183). C. P. Davis-Stober, A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths, Journal of Mathematical Psychology, 54, 471-474. M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique., Ph.D Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy) Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019. Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011. See Theorem 8.4. E. Fusy, D. Poulalhon, and G. Schaeffer, Bijective counting of plane bipolar orientations and Schnyder words, Eur. J. Combinat. 30 (2009) 1646-1658, eq (2). Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, and Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017. D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer 1986. D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy) D. Gouyou-Beauchamps. Standard Young tableaux of height 4 and 5, European J. Combin. 10 (1989) 69 - 82. Nicholas M. Katz, A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers, preprint, 2015; Mathemtika, Volume 62, Issue 3 2016 , pp. 811-817. Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010. Alec Mihailovs, Enumeration of walks on lattices, arXiv:math/9803128 (1998). 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. Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019. Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021). Index entries for sequences related to Young tableaux. FORMULA G.f.: 3F2( [ 1, 1/2, 3/2 ]; [ 3, 4 ]; 16 x ). a(n) = 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!) (Mihailovs). a(n) = Det[Table[binomial[i+1, j-i+2], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005 a(n) = b(n)b(n+1)/6 where b(n) is the superballot number A007054. - David Callan, Feb 01 2007 a(n) = A000108(n)*A000108(n+2) - A000108(n+1)^2. - Philippe Deléham, Apr 11 2007 G.f.: (1 + 6*x - hypergeom([-1/2,-3/2],,16*x))/(4*x^2). - Mark van Hoeij, Nov 02 2009 From Michael Somos, Oct 02 2014: (Start) a(n) = 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!). D-finite with recurrence 0 = a(n) * 4*(2*n+1)*(2*n+3) - a(n+1) * (n+3)*(n+4) for all n in Z. 0 = a(n)*(+65536*a(n+2) - 72192*a(n+3) + 10296*a(n+4)) + a(n+1)*(-1536*a(n+2) - 1632*a(n+3) - 282*a(n+4)) + a(n+2)*(+40*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z. 0 = a(n)^2*a(n+2)*(+1792*a(n+1) - 882*a(n+2)) + a(n)*a(n+1)^2*(+768*a(n+1) + 580*a(n+2)) + 7*a(n)*a(n+1)*a(n+2)^2 +a(n+1)^3*(-18*a(n+1) + 3*a(n+2)) for all n in Z. (End) a(n) ~ 3 * 2^(4*n+3) / (Pi * n^5). - Vaclav Kotesovec, Feb 10 2015 From Peter Bala, Feb 22 2023: (Start) a(n) = (12*(2*n - 1)/((n + 1)(n + 2)(n + 3))) * Catalan(n-1)*Catalan(n+1) for n >= 1. a(n) = Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j). a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j - 1) for n >= 1. (End) Sum_{n>=0} a(n)/16^n = 88 - 4096/(15*Pi). - Amiram Eldar, May 06 2023 EXAMPLE Example: a(2)=3 counts EWEW, EEWW, ENSW. G.f. = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 594*x^5 + 4719*x^6 + 40898*x^7 + ... MATHEMATICA CoefficientList[ Series[ HypergeometricPFQ[ {1, 1/2, 3/2}, {3, 4}, 16 x], {x, 0, 19}], x] a[ n_] := If[ n < 1, Boole[n == 0], Det[ Table[ Binomial[i + 1, j - i + 2], {i, n}, {j, n}]]]; (* Michael Somos, Feb 25 2014 *) (* slight modification of David Callan formula *) a[ n_] := 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!); (* Michael Somos, Oct 02 2014 *) PROG (Magma) [6*Factorial(2*n)*Factorial(2*n+2)/(Factorial(n)*Factorial(n+1)* Factorial(n+2)*Factorial(n+3)): n in [0..25]]; // Vincenzo Librandi, Aug 04 2011 (PARI) a(n)=6*binomial(2*n+2, n)*(2*n)!/(n+1)!/(n+3)! \\ Charles R Greathouse IV, Aug 04 2011 (LiE) p_tensor(2*n, [0, 1], B2)|[0, 0] (LiE) p_tensor(2*n, [1, 0], C2)|[0, 0] (PARI) {a(n) = if( n<0, if( n<-2, 0, [-3/2, -1/4][-n]), 6 * (2*n)! * (2*n+2)! / (n! * (n+1)! * (n+2)! * (n+3)!))}; /* Michael Somos, Oct 02 2014 */ CROSSREFS A column of the triangle in A179898. A diagonal of the triangle in A185249. Row sums of A193691, A193692. - Alois P. Heinz, Aug 03 2011 See A138349 for another version. Cf. A000108, A006149, A006150, A248152. Sequence in context: A256337 A256330 A190761 * A220911 A088717 A111538 Adjacent sequences: A005697 A005698 A005699 * A005701 A005702 A005703 KEYWORD nonn,walk,easy AUTHOR N. J. A. Sloane EXTENSIONS More terms from James A. Sellers, Dec 24 1999 Corrected by Vladeta Jovovic, May 23 2004 Better definition from David Callan, Sep 18 2007 Definition simplified by N. J. A. Sloane, Nov 30 2020 STATUS approved

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