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A005568 Product of two successive Catalan numbers C(n)*C(n+1).
(Formerly M1972)
17
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; text; internal format)
OFFSET

0,2

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_{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, 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

T. D. Noe, Table of n, a(n) for n=0..100

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

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.

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

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 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, Europ. J. Combinatorics, 10, 1989, 69-82.

D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

R. K. Guy, Letter to N. J. A. Sloane, May 1990

R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), #00.1.6.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

G. 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.

R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.

R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy]

Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005-2006.

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).

(n+1)*n*a(n) = 4*(2*n-1)*(2*n-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))). - 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

MAPLE

c:=n->binomial(2*n, n)/(n+1): seq(c(n)*c(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 *)

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 */

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

CROSSREFS

Cf. A000108, A000356, A005817.

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

Sequence in context: A166076 A217938 A051405 * A036075 A212914 A123881

Adjacent sequences:  A005565 A005566 A005567 * A005569 A005570 A005571

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from Emeric Deutsch, Feb 20 2004

More terms from Manuel Kauers, Nov 18 2008

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

STATUS

approved

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Last modified February 24 03:49 EST 2018. Contains 299595 sequences. (Running on oeis4.)