cp's OEIS Frontend

A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2.

%I A000891 #168 Jun 30 2025 10:17:21
%S A000891 1,3,20,175,1764,19404,226512,2760615,34763300,449141836,5924217936,
%T A000891 79483257308,1081724803600,14901311070000,207426250094400,
%U A000891 2913690606794775,41255439318353700,588272005095043500
%N A000891 a(n) = (2*n)!*(2*n+1)! / (n! * (n+1)!)^2.
%C A000891 Number of parallelogram polyominoes having n+1 columns and n+1 rows. - _Emeric Deutsch_, May 21 2003
%C A000891 Number of tilings of an <n,2,n> hexagon.
%C A000891 a(n) is the 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
%C A000891 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
%C A000891 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
%C A000891 The number of proper mergings of two n-chains. - _Henri Mühle_, Aug 17 2012
%C A000891 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
%C A000891 Also the number of ordered rooted trees with 2(n+1) nodes and n+1 leaves, i.e., half of the nodes are leaves. These trees are ranked by A358579. The unordered version is A185650. - _Gus Wiseman_, Nov 27 2022
%C A000891 The number of secondary GL(2) invariants constructed from n+1 two component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - _Jaco van Zyl_, Jun 30 2025
%D A000891 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
%D A000891 E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.
%H A000891 Vincenzo Librandi, <a href="/A000891/b000891.txt">Table of n, a(n) for n = 0..100</a>
%H A000891 Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, <a href="https://arxiv.org/abs/2105.00971">Enumeration of parallelogram polycubes</a>, arXiv:2105.00971 [cs.DM], 2021.
%H A000891 Elena Barcucci, Andrea Frosini and Simone Rinaldi, <a href="http://dx.doi.org/10.1016/j.disc.2005.01.006">On directed-convex polyominoes in a rectangle</a>, Discr. Math., 298 (2005). 62-78.
%H A000891 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
%H A000891 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011), Article 11.4.5.
%H A000891 William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, <a href="http://arxiv.org/abs/0804.2930">Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions</a>, arXiv:0804.2930 [math.CO], 2008.
%H A000891 William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, <a href="http://dx.doi.org/10.1016/j.disc.2008.06.042">Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions</a>, Discrete Math., 309 (2009), 2834-2838.
%H A000891 Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, <a href="https://arxiv.org/abs/2504.14181">Bosonic Fortuity in Vector Models</a>, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
%H A000891 Ivan Marin and Emmanuel Wagner, <a href="http://arxiv.org/abs/1203.5981">A cubic defining algebra for the Links-Gould polynomial</a>. arXiv preprint arXiv:1203.5981 [math.GT], 2012. - From _N. J. A. Sloane_, Sep 21 2012
%H A000891 Henri Mühle, <a href="http://arxiv.org/abs/1206.3922">Counting Proper Mergings of Chains and Antichains</a>, arXiv:1206.3922 [math.CO], 2012.
%H A000891 Guoce Xin, <a href="http://dx.doi.org/10.1016/j.aam.2007.04.007">Determinant formulas relating to tableaux of bounded height</a>, Adv. Appl. Math. 45 (2010) 197-211.
%F A000891 -4*a(n) = A010370(n+1).
%F A000891 G.f.: (1 - E(16*x)/(Pi/2))/(4*x) where E() is the elliptic integral of the second kind. [edited by _Olivier Gérard_, Feb 16 2011]
%F A000891 G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x)) / (4*x). - _Olivier Gérard_, Feb 16 2011
%F A000891 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
%F A000891 a(n) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - _Zerinvary Lajos_, Jun 06 2007
%F A000891 For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - _Gary W. Adamson_, Apr 08 2011
%F A000891 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).
%F A000891 G.f.: If G_N(x) = 1 + Sum_{k=1..N} ((2*k)!*(2*k+1)!*x^k)/(k!*(k+1)!)^2, 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
%F A000891 D-finite with recurrence (n+1)^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012
%F A000891 a(n) = A005558(2n). - _Mark van Hoeij_, Aug 20 2014
%F A000891 a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - _Reinhard Zumkeller_, Sep 30 2014
%F A000891 From _Ilya Gutkovskiy_, Feb 01 2017: (Start)
%F A000891 E.g.f.: 2F2(1/2,3/2; 2,2; 16*x).
%F A000891 a(n) ~ 2^(4*n+1)/(Pi*n^2). (End)
%F A000891 a(n) = A005408(n)*(A000108(n))^2. - _Ivan N. Ianakiev_, Nov 13 2019
%F A000891 a(n) = det(M(n)) where M(n) is the n X n matrix with m(i,j) = binomial(n+j+1,i+1). - _Benoit Cloitre_, Oct 22 2022
%F A000891 a(n) = Integral_{x=0..16} x^n*W(x) dx, where W(x) = (16*EllipticE(1 - x/16) - x*EllipticK(1 - x/16))/(8*Pi^2*sqrt(x)), n=>0. W(x) diverges at x=0, monotonically decreases for x>0, and vanishes at x=16. EllipticE and EllipticK are elliptic functions. This integral representation as n-th moment of a positive function W(x) on the interval [0, 16] is unique. - _Karol A. Penson_, Dec 20 2023
%e A000891 G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
%e A000891 From _Gus Wiseman_, Nov 27 2022: (Start)
%e A000891 The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
%e A000891   (((o)oo))  (((o)o)o)  (((o))oo)
%e A000891   (((oo)o))  (((oo))o)  ((o)(o)o)
%e A000891   (((ooo)))  ((o)(oo))  ((o)o(o))
%e A000891   ((o(o)o))  ((o(o))o)  (o((o))o)
%e A000891   ((o(oo)))  ((oo)(o))  (o(o)(o))
%e A000891   ((oo(o)))  (o((o)o))  (oo((o)))
%e A000891              (o((oo)))
%e A000891              (o(o(o)))
%e A000891 (End)
%p A000891 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
%t A000891 a[ n_] := If[ n == -1, 0, Binomial[2 n + 1, n]^2 / (2 n + 1)]; (* _Michael Somos_, May 28 2014 *)
%t A000891 a[ n_] := SeriesCoefficient[ (1 - Hypergeometric2F1[ -1/2, 1/2, 1, 16 x]) / (4 x), {x, 0, n}]; (* _Michael Somos_, May 28 2014 *)
%t A000891 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 *)
%t A000891 a[ n_] := SeriesCoefficient[ (1 - EllipticE[ 16 x] / (Pi/2)) / (4 x), {x, 0, n}]; (* _Michael Somos_, Sep 18 2016 *)
%t A000891 a[n_] := (2 n + 1) CatalanNumber[n]^2;
%t A000891 Array[a, 20, 0] (* _Peter Luschny_, Mar 03 2020 *)
%o A000891 (PARI) {a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* _Michael Somos_, Jun 22 2005 */
%o A000891 (PARI) a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ _Hugo Pfoertner_, Oct 22 2022
%o A000891 (Magma) [Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // _Vincenzo Librandi_, Aug 15 2011
%o A000891 (Haskell)
%o A000891 a000891 n = a001263 (2 * n - 1) n  -- _Reinhard Zumkeller_, Oct 10 2013
%Y A000891 Cf. A000356, A010370, A038535.
%Y A000891 Cf. A145600, A145601, A145602, A145603. - _Peter Bala_, Oct 14 2008
%Y A000891 Cf. A000142, A000894, A248045.
%Y A000891 Equals half of A267981.
%Y A000891 Counts the trees ranked by A358579.
%Y A000891 A001263 counts ordered rooted trees by nodes and leaves.
%Y A000891 A090181 counts ordered rooted trees by nodes and internals.
%Y A000891 Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.
%K A000891 nonn
%O A000891 0,2
%A A000891 _N. J. A. Sloane_
%E A000891 More terms from _Andrew V. Sutherland_, Mar 24 2008