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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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23
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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S. Dulucq and O. Guibert, Baxter permutations, Discrete Math., Vol. 180, No. 1-3 (1998), pp. 143--156. MR1603713 (99c:05004). See Th. 6.
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FORMULA
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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
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.
(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
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MAPLE
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A000108:=n->binomial(2*n, n)/(n+1):
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Two hypergeometric g.f.s, van Hoeij's formula checked and formula field edited by Olivier Gérard, Feb 16 2011
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STATUS
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approved
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