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A005568 Product of two successive Catalan numbers C(n)*C(n+1).
(Formerly M1972)

%I M1972

%S 1,2,10,70,588,5544,56628,613470,6952660,81662152,987369656,

%T 12228193432,154532114800,1986841476000,25928281261800,

%U 342787130211150,4583937702039300,61923368957373000,844113292629453000,11600528392993339800,160599522947154548400,2238236829690383152800

%N Product of two successive Catalan numbers C(n)*C(n+1).

%C Also equal to the number of standard tableaux of 2n cells with height less than or equal to 4. A005817(2n) - _Mike Zabrocki_, Feb 22 2007

%C Also equal to Sum binomial(2n,2i)*C(i)*C(n-i) = (4/Pi^2) Integral_{y=0..Pi} Integral_{x=0..Pi} (2*cos(x)+2*cos(y))^(2n)*sin^2(x)*sin^2(y) dx dy, since this counts walks of 2n steps in the nonnegative quadrant of an integer lattice that return to the origin (cf. R. K. Guy link below). - _Andrew V. Sutherland_, Nov 29 2007

%C Also, number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, 0), (-1, 1), (1, -1), (1, 0)}. - _Manuel Kauers_, Nov 18 2008 - _Manuel Kauers_, Nov 18 2008

%C Also, number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, 0), (0, -1), (0, 1), (1, 0)}. - _Manuel Kauers_, Nov 18 2008

%C a(2n-2) is also the sum of the numbers of standard Young tableaux of size 2n of (2,2) rectangular hook shapes (k+2,k+2,2^{n-2-k}, 0 <= k <= n-2. - Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010

%C Also, number of tree-rooted planar maps with n edges. - _Noam Zeilberger_, Aug 18 2017

%D M. Lothaire, Applied Combinatorics on Words, Cambridge, 2005. See Prop. 9.1.9, p. 452. [From _N. J. A. Sloane_, Apr 03 2012]

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

%H Muniru A Asiru, <a href="/A005568/b005568.txt">Table of n, a(n) for n = 0..831</a> (terms n=0..100 from T. D. Noe)

%H M. Agiorgousis, B. Green, N. A. Scoville, A. Onderdonk and K. Rich, <a href="http://webpages.ursinus.edu/nscoville/DMFIntegersSubmission.pdf">Homological sequences in discrete Morse theory</a>, J. Integer Seqs., (submitted), 2012. - From _N. J. A. Sloane_, Dec 27 2012

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H Matteo Bellitti, Siddhardh Morampudi, Chris R. Laumann, <a href="https://arxiv.org/abs/1908.02263">Hamiltonian Dynamics of a Sum of Interacting Random Matrices</a>, arXiv:1908.02263 [cond-mat.stat-mech], 2019.

%H Olivier Bernardi, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1r9">Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems</a>, Electronic Journal of Combinatorics, Vol. 14 (2007), R9.

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008.

%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%H Steve Butler et al., <a href="http://www.math.ucsd.edu/~ronspubs/pre_paperclip.pdf">Paperclip graphs</a>, see p. 10.

%H R. Cori et al., <a href="http://dx.doi.org/10.1016/0097-3165(86)90018-X">Shuffle of parenthesis systems and Baxter permutations</a>, J. Combin. Theory, A 43 (1986), 1-22.

%H D. Gouyou-Beauchamps, <a href="/A005700/a005700.pdf">Chemins sous-diagonaux et tableaux de Young</a>, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)

%H D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1016/S0195-6698(89)80034-4">Standard Young tableaux of height 4 and 5</a>, Europ. J. Combinatorics, 10, 1989, 69-82.

%H D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1007/BFb0072513">Chemins sous-diagonaux et tableaux de Young</a>, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>

%H R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.6.

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.

%H G. Lachaud, <a href="http://arxiv.org/abs/1506.06482">On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius</a>, arXiv preprint arXiv:1506.06482 [math.AG], 2015.

%H James Mallos, <a href="https://doi.org/10.3390/math7020165">A 6-Letter 'DNA' for Baskets with Handles</a>, Mathematics (2019) Vol. 7, No. 2, 165.

%H R. C. Mullin, <a href="http://dx.doi.org/10.1016/S0021-9800(67)80001-2">On the average activity of a spanning tree of a rooted map</a>, J. Combin. Theory, 3 (1967), 103-121.

%H R. C. Mullin, <a href="/A260039/a260039.pdf">On the average activity of a spanning tree of a rooted map</a>, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy]

%H Liviu I. Nicolaescu, <a href="http://arxiv.org/abs/math/0512496">Counting Morse functions on the 2-sphere</a>, arXiv:math/0512496 [math.GT], 2005-2006.

%H Dov Tamari, <a href="https://doi.org/10.24033/bsmf.1446">Monoïdes préordonnés et chaînes de Malcev</a>, Bulletin de la Société Mathématique de France, Volume 82 (1954), 53-96. See end of Appendix II.

%H Walsh, T. R. S.; Lehman, A. B.; <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259: the number of parenthesis-bracket systems with n pairs.

%F a(n) = binomial(2*n,n)*binomial(2*n+2,n+1)/((n+1)(n+2)).

%F a(n) = 2*(2*n+1)*binomial(2*n,n)^2/((n+2)(n+1)^2).

%F D-finite with recurrence (n+2)*(n+1)*a(n) = 4*(2*n-1)*(2*n+1)*a(n-1). - Corrected _R. J. Mathar_, Feb 05 2020

%F G.f. in Maple notation: (1/2)/x+1/768/(x^2*Pi)*((32-512*x)*EllipticK(4*x^(1/2))+(-32-512*x)*EllipticE(4*x^(1/2))). - _Karol A. Penson_, Oct 24 2003

%F G.f.: 3F2( (1, 1/2, 3/2); (2, 3))(16*x) = (1 - 2F1((-1/2, 1/2); (2))( 16*x))/(2*x). - _Olivier Gérard_ Feb 16 2011

%F G.f.: (1/(6*x))*(3+(16*x-1)*(2*hypergeom([1/2, 1/2],[1],16*x) + (16*x+1)*hypergeom([3/2, 3/2],[2],16*x))). - _Mark van Hoeij_, Nov 02 2009

%F G.f.: (1-hypergeom([-1/2,1/2],[2],16*x))/(2*x). - _Mark van Hoeij_, Aug 14 2014

%F E.g.f.: (1/3)*(8*x^2*BesselI(0, 2*x)^2 - 4*BesselI(0, 2*x)*BesselI(1, 2*x)*x - BesselI(1, 2*x)^2 - 8*BesselI(1, 2*x)^2*x^2)/x. - _Vladeta Jovovic_, Dec 29 2003

%F E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)^2/x^2. - _Michael Somos_, Jun 22 2005

%F From _Paul D. Hanna_, Nov 26 2009: (Start)

%F G.f.: A(x) = [(1/x)*Series_Reversion(x/F(x)^2)]^(1/2) where F(x) = g.f. of A004304, where A004304(n) is the number of nonseparable planar tree-rooted maps with n edges.

%F G.f.: A(x) = F(x*A(x)^2) where A(x/F(x)^2) = F(x) where F(x) = g.f. of A004304.

%F G.f.: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) where G(x) = g.f. of A168450.

%F G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A168450.

%F Self-convolution yields A168452.

%F (End)

%F Representation of a(n) as the n-th power moment of a positive function on the segment [0,16]; in Mathematica notation, a(n) = NIntegrate[x^n*(8 ((1+x/16)*EllipticE[1-x/16]-1/8*x*EllipticK[1-x/16]))/(3*(Pi^2)*Sqrt[x]),{x,0,16}]. This solution of the Hausdorff power moment problem is unique. - _Karol A. Penson_, Oct 05 2011

%F G.f. y=A(x) satisfies: 0 = x^2*(16*x-1)*y''' + 6*x*(16*x-1)*y'' + 6*(18*x-1)*y' + 12*y. - _Gheorghe Coserea_, Jun 14 2018

%p A000108:=n->binomial(2*n,n)/(n+1):

%p seq(A000108(n)*A000108(n+1),n=0..21); # _Emeric Deutsch_, Mar 05 2007

%t f[n_] := CatalanNumber[n] CatalanNumber[n + 1] (* Or *) (4n + 2) Binomial[2 n, n]^2/(n^3 + 4n^2 +5n + 2) (* Or *) (2 n)! (2 + 2 n)!/(n! ((1 + n)!)^2 (2 + n)!); Array[f, 22, 0] (* _Robert G. Wilson v_ *)

%t Times@@@Partition[CatalanNumber[Range[0,30]],2,1] (* _Harvey P. Dale_, Jul 23 2012 *)

%o (PARI) (alias(C,binomial));a(n)=(C(2*n,n)-C(2*n,n-1))*(C(2*n+2,n+1)-C(2*n+2,n)) /* _Michael Somos_, Jun 22 2005 */

%o (Sage) [catalan_number(i)*catalan_number(i+1) for i in range(0,22)] # _Zerinvary Lajos_, May 17 2009

%o (GAP) List([0..21],n->Binomial(2*n,n)*Binomial(2*(n+1),n+1)/((n+1)*(n+2))); # _Muniru A Asiru_, Dec 13 2018

%o (MAGMA) [Catalan(n)*Catalan(n+1): n in [0..21]]; // _Vincenzo Librandi_, Feb 06 2020

%Y Cf. A000108, A000356, A005817.

%Y Cf. A004304, A168450, A168451, A168452. - _Paul D. Hanna_, Nov 26 2009

%K nonn,easy

%O 0,2

%A _R. K. Guy_, _Simon Plouffe_ and _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Feb 20 2004

%E More terms from _Manuel Kauers_, Nov 18 2008

%E Two hypergeometric gfs, van Hoeij's formula checked and formula field edited by _Olivier Gérard_, Feb 16 2011

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Last modified September 23 22:32 EDT 2020. Contains 337315 sequences. (Running on oeis4.)