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A005568
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Product of two successive Catalan numbers C(n)*C(n+1).
(Formerly M1972)
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16
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1, 2, 10, 70, 588, 5544, 56628, 613470, 6952660, 81662152, 987369656, 12228193432, 154532114800, 1986841476000, 25928281261800, 342787130211150, 4583937702039300, 61923368957373000, 844113292629453000, 11600528392993339800, 160599522947154548400, 2238236829690383152800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also equal to the number of standard tableaux of 2n cells with height less than or equal to 4. A005817(2n) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 22 2007
Also equal to Sum binomial(2n,2i)C(i)C(n-i) = (4/pi^2) Integral_{0 .. pi} Integral_{0^pi} (2cos(x)+2cos(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 (drew(AT)math.mit.edu), 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 (manuel(AT)kauers.de), Nov 18 2008 - Manuel Kauers (manuel(AT)kauers.de), 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 (manuel(AT)kauers.de), Nov 18 2008 - Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
Contribution from Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010: (Start)
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. (End)
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REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Olivier Bernardi, Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems, Electronic Journal of Combinatorics, Vol. 14 (2007), R9.
R. Cori et al., Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory, A 43 (1986), 1-22.
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
D. Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, Europ. J. Combinatorics, 10, 1989, 69-82.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.
Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496.
A. Regev, Preprint. [From Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
M. Bousquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
| a(n)=binomial(2n, n)*binomial(2n+2, n+1)/[(n+1)(n+2)] = 2(2n+1)binomial(2n, n)^2/[(n+2)(n+1)^2].
(n+1)*n*a(n) = 4*(2n-1)*(2n-3)*a(n-1).
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))); from Karol A. Penson (penson(AT)lptl.jussieu.fr), 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) [From Olivier Gerard (olivier.gerard(AT)gmail.com) 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))) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 02 2009]
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 (vladeta(AT)eunet.rs), 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
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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 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. - From Karol A. Penson, Oct 05 2011
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MAPLE
| c:=n->binomial(2*n, n)/(n+1): seq(c(n)*c(n+1), n=0..21); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 05 2007
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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] (* RGWv *)
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PROG
| (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 */
(Other) sage: [catalan_number(i)*catalan_number(i+1) for i in xrange(0, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
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CROSSREFS
| Cf. A000108.
Cf. A000356.
Cf. A005817.
Cf. A004304, A168450, A168451, A168452. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009]
Sequence in context: A121201 A166076 A051405 * A036075 A123881 A089845
Adjacent sequences: A005565 A005566 A005567 * A005569 A005570 A005571
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KEYWORD
| nonn,easy
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AUTHOR
| R. K. Guy, Simon Plouffe, N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 20 2004
More terms from Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008
Two hypergeometric GFs, van Hoeij's formula checked and formula field edited by Olivier Gérard (olivier.gerard(AT)gmail.com), Feb 16 2011
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