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

1, 2, 10, 70, 588, 5544, 56628, 613470, 6952660, 81662152, 987369656, 12228193432, 154532114800, 1986841476000, 25928281261800, 342787130211150, 4583937702039300, 61923368957373000, 844113292629453000, 11600528392993339800, 160599522947154548400, 2238236829690383152800

Offset: 0

Author

R. K. Guy, Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

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

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

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

Also, number of tree-rooted planar maps with n edges. - Noam Zeilberger, Aug 18 2017

References

• M. Lothaire, Applied Combinatorics on Words, Cambridge, 2005. See Prop. 9.1.9, p. 452. [From N. J. A. Sloane, Apr 03 2012]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..831 (terms n=0..100 from T. D. Noe)
• M. Agiorgousis, B. Green, N. A. Scoville, A. Onderdonk, and K. Rich, Homological sequences in discrete Morse theory, J. Integer Seqs., (submitted), 2012. - From _N. J. A. Sloane_, Dec 27 2012
• Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
• Matteo Bellitti, Siddhardh Morampudi, and Chris R. Laumann, Hamiltonian Dynamics of a Sum of Interacting Random Matrices, arXiv:1908.02263 [cond-mat.stat-mech], 2019.
• Olivier Bernardi, Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems, Electronic Journal of Combinatorics, Vol. 14 (2007), Article R9.
• Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.
• Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
• Steve Butler et al., Paperclip graphs, see p. 10.
• Robert Cori, Serge Dulucq and Gérard Viennot, Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory, Series A, Vol. 43, No. 1 (1986), pp. 1-22.
• Serge Dulucq and Olivier Guibert, Baxter permutations, Discrete Math., Vol. 180, No. 1-3 (1998), pp. 143--156. MR1603713 (99c:05004). See Th. 6.
• Dominique 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)
• Dominique Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, Europ. J. Combinatorics, Vol. 10, No. 1 (1989), pp. 69-82.
• Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986, p. 118 Corr. a).
• Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
• Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
• Gilles Lachaud, On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius, arXiv preprint arXiv:1506.06482 [math.AG], 2015.
• Cécilia Lancien, Patrick Oliveira Santos, and Pierre Youssef, Central Limit Theorem for tensor products of free variables, arXiv:2404.19662 [math.PR], 2024. See p. 7.
• James Mallos, A 6-Letter 'DNA' for Baskets with Handles, Mathematics, Vol. 7, No. 2 (2019), Article 165.
• Ronald C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, Vol. 3, No. 2 (1967), pp. 103-121.
• Ronald C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, Vol. 3, No. 2 (1967), pp. 103-121. [Annotated scanned copy]
• Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005-2006.
• Justin Offutt, Automatic Bounds on Constant Term Sequences Modulo Primes, arXiv:2504.19031 [math.NT], 2025. See p. 2.
• Dov Tamari, Monoïdes préordonnés et chaînes de Malcev, Bulletin de la Société Mathématique de France, Volume 82 (1954), pp. 53-96. See end of Appendix II.
• Timothy R. S. Walsh and Alfred B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory, Ser. B, Vol. 18, No. 3 (1975), pp. 222-259: the number of parenthesis-bracket systems with n pairs.

Crossrefs

Cf. A000108, A000356, A005817.

Cf. A004304, A168450, A168451, A168452. - Paul D. Hanna, Nov 26 2009

Programs

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

Magma

[Catalan(n)*Catalan(n+1): n in [0..21]]; // Vincenzo Librandi, Feb 06 2020

Maple

A000108:=n->binomial(2*n,n)/(n+1):
seq(A000108(n)*A000108(n+1),n=0..21); # Emeric Deutsch, Mar 05 2007

Mathematica

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 *)
Times@@@Partition[CatalanNumber[Range[0,30]],2,1] (* Harvey P. Dale, Jul 23 2012 *)

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 */

Sage

[catalan_number(i)*catalan_number(i+1) for i in range(0,22)] # Zerinvary Lajos, May 17 2009

Formula

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

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

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

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

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

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

G.f.: (1-hypergeom([-1/2,1/2],[2],16*x))/(2*x). - Mark van Hoeij, Aug 14 2014

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

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

From Paul D. Hanna, Nov 26 2009: (Start)

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.

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

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

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

Self-convolution yields A168452.

(End)

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

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

Sum_{n>=0} a(n)/4^(2*n+1) = 2 - 16/(3*Pi). - Amiram Eldar, Apr 02 2022

Extensions

More terms from Emeric Deutsch, Feb 20 2004

More terms from Manuel Kauers, Nov 18 2008

Two hypergeometric g.f.s, van Hoeij's formula checked and formula field edited by Olivier Gérard, Feb 16 2011