0,2

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

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

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.

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

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.

M. Bouquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

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.

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

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]

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

(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]

Cf. A000108.

Cf. A005817.

Sequence in context: A121201 A166076 A051405 this_sequence A036075 A123881 A089845

Adjacent sequences: A005565 A005566 A005567 this_sequence A005569 A005570 A005571

nonn,easy,new

R. K. Guy, Simon Plouffe, N. J. A. Sloane (njas(AT)research.att.com).

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