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 A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2. 21
 1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003 Number of tilings of a hexagon. a(n) = number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan, Jul 25 2005 a(n)=A001700(n)*A000108(n) =(1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007 The number of returning walks of length 2n on the upper half of a square lattice, since a(n)=Sum_{k=0..2n}Binomial(2n,k)A126120(k)A126869(n-k). - Andrew V. Sutherland, Mar 24 2008 For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). - Peter Bala, Oct 14 2008 For n>0, A000891(n)/(n+2) = A000356 starting (1, 5, 35, 294,...). - Gary W. Adamson, Apr 08 2011 The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012 a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014 a(n) is number of pairs of non-intersecting lattice paths from (0,0) to (n+1,n+1) using (1,0) and (0,1) as steps. Here, non-intersecting means two paths do not share a vertex except the origin and the destination. For example, a(1) = 3 because we have three such pairs from (0,0) to (2,2): {NNEE,EENN}, {NNEE,ENEN}, {NENE,EENN}. - Ran Pan, Oct 01 2015 REFERENCES J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8. E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78. Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4. P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5 W. Y. C. Chen, S. X. M. Pang, E. X. Y. Qu and R. P Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008. W. Y. C. Chen, S. X. M. Pang, E. X. Y. Qu and R. P Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838. I. Marin and E. Wagner, A cubic defining algebra for the Links-Gould polynomial. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From N. J. A. Sloane, Sep 21 2012 H. Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012. FORMULA G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x) * BesselI(1, 2*x) / x. - Michael Somos, Jun 22 2005 a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle). G.f.: If G_N(x)=1+sum('((2*k)!*(2*k+1)!*x^k)/(((k!)*((k+1)!))^2)', 'k'=1..N), G_N(x)=1+12*x/(G(0)-12*x); G(k)=16*x*(k^2)+32*x*k+(k^2)+4*k+12*x+4-4*x*(2*k+3)*(2*k+5)*((k+2)^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011 (n+1)^2*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Dec 03 2012 a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014 -4 * a(n) = A010370(n+1). From Ilya Gutkovskiy, Feb 01 2017: (Start) E.g.f.: 2F2(1/2,3/2; 2,2; 16*x). a(n) ~ 2^(4*n+1)/(Pi*n^2). (End) EXAMPLE G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ... MAPLE with(combstruct): bin := {B=Union(Z, Prod(B, B))} :seq(1/2*binomial(2*i, i)*(count([B, bin, unlabeled], size=i)), i=1..18) ; # Zerinvary Lajos, Jun 06 2007 MATHEMATICA a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* Michael Somos, May 28 2014 *) a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* Michael Somos, May 28 2014 *) a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ BesselI[0, 2 x] BesselI[1, 2 x] / x, {x, 0, 2 n}]]; (* Michael Somos, May 28 2014 *) a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* Michael Somos, Sep 18 2016 *) PROG (PARI) {a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */ (MAGMA) [Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011 (Haskell) a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013 CROSSREFS Cf. A000356, A010370, A038535. Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008 Cf. A000142, A000894, A248045. Sequence in context: A213377 A216583 A154644 * A242164 A129840 A085390 Adjacent sequences:  A000888 A000889 A000890 * A000892 A000893 A000894 KEYWORD nonn AUTHOR EXTENSIONS More terms from Andrew V. Sutherland, Mar 24 2008 Legend for G.f. formula plus another hypergeometric variation from Olivier Gérard, Feb 16 2011 STATUS approved

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