A145600 -id:A145600 - cp's OEIS frontend

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

1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500

Offset: 0

N. J. A. Sloane

nonn

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, May 21 2003
Number of tilings of an  hexagon.
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
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
The number of proper mergings of two n-chains. - Henri Mühle, Aug 17 2012
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
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
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

			G.f. = 1 + 3*x + 20*x^2 + 175*x^3 + 1764*x^4 + 19404*x^5 + ...
From _Gus Wiseman_, Nov 27 2022: (Start)
The a(2) = 20 ordered rooted trees with 6 nodes and 3 leaves:
  (((o)oo))  (((o)o)o)  (((o))oo)
  (((oo)o))  (((oo))o)  ((o)(o)o)
  (((ooo)))  ((o)(oo))  ((o)o(o))
  ((o(o)o))  ((o(o))o)  (o((o))o)
  ((o(oo)))  ((oo)(o))  (o(o)(o))
  ((oo(o)))  (o((o)o))  (oo((o)))
             (o((oo)))
             (o(o(o)))
(End)

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
Elena Barcucci, Andrea Frosini and Simone 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.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, arXiv:0804.2930 [math.CO], 2008.
William Y. C. Chen, Sabrina X. M. Pang, Ellen X. Y. Qu, and Richard P. Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, Discrete Math., 309 (2009), 2834-2838.
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Ivan Marin and Emmanuel 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
Henri Mühle, Counting Proper Mergings of Chains and Antichains, arXiv:1206.3922 [math.CO], 2012.
Guoce Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A000356, A010370, A038535.
Cf. A145600, A145601, A145602, A145603. - Peter Bala, Oct 14 2008
Cf. A000142, A000894, A248045.
Equals half of A267981.
Counts the trees ranked by A358579.
A001263 counts ordered rooted trees by nodes and leaves.
A090181 counts ordered rooted trees by nodes and internals.
Cf. A000108, A065097, A080936, A185650, A358371, A358586, A358590.

a000891 n = a001263 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 10 2013

[Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

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

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 *)
a[n_] := (2 n + 1) CatalanNumber[n]^2;
Array[a, 20, 0] (* Peter Luschny, Mar 03 2020 *)

{a(n) = binomial(2*n+1, n)^2 / (2*n + 1)}; /* Michael Somos, Jun 22 2005 */

a(n) = matdet(matrix(n, n, i, j, binomial(n+j+1,i+1))) \\ Hugo Pfoertner, Oct 22 2022

-4*a(n) = A010370(n+1).
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]
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
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) = A001700(n)*A000108(n) = (1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos, Jun 06 2007
For n > 0, a(n) = (n+2)*A000356(n) starting (1, 5, 35, 294, ...). - Gary W. Adamson, Apr 08 2011
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_{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
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
a(n) = A005558(2n). - Mark van Hoeij, Aug 20 2014
a(n) = A000894(n) / (n+1) = A248045(n+1) / A000142(n+1). - Reinhard Zumkeller, Sep 30 2014
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)
a(n) = A005408(n)*(A000108(n))^2. - Ivan N. Ianakiev, Nov 13 2019
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
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

More terms from Andrew V. Sutherland, Mar 24 2008

A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2binomial(n + 1, k + 1)binomial(n + 1, k - 1)/(n + 1).

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 20, 20, 4, 5, 40, 75, 40, 5, 6, 70, 210, 210, 70, 6, 7, 112, 490, 784, 490, 112, 7, 8, 168, 1008, 2352, 2352, 1008, 168, 8, 9, 240, 1890, 6048, 8820, 6048, 1890, 240, 9, 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10

Offset: 1

Peter Bala, Oct 14 2008

T(n,k) is the number of walks of n unit steps on the square lattice (i.e., each step in the direction either up (U), down (D), right (R) or left (L)) starting from (0,0) and finishing at points on the horizontal line y = 1, which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 1 of the generalized Narayana numbers N_r(n, k) := (r + 1)/(n + 1)*binomial(n + 1, k + r)*binomial(n + 1, k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with row numbering starting at n = 0). For other cases see A145597 (r = 2), A145598 (r = 3) and A145599 (r = 4).
T(n,k) is the number of preimages of the permutation 21345...(n+2) under West's stack-sorting map that have exactly k descents. - Colin Defant, Sep 15 2018
T(n+k+1,k+1) equals the number of tilings of an octagon with internal angles of 135 degrees and sides of lengths n, k, 1, 1, n, k, 1, 1 using unit rhombi with internal angles 45 degrees and 135 degrees. See Elnitzky, Theorem 5.1. - Peter Bala, Apr 25 2022

			n\k|..1.....2....3.....4.....5.....6
====================================
.1.|..1
.2.|..2.....2
.3.|..3.....8....3
.4.|..4....20...20.....4
.5.|..5....40...75....40.....5
.6.|..6....70..210...210....70.....6
...
Row 3 entries:
T(3,1) = 3: the 3 walks from (0,0) to (-2,1) of three steps are LLU, LUL and ULL.
T(3,2) = 8: the 8 walks from (0,0) to (0,1) of three steps are UDU, UUD, ULR, URL, RLU, LRU, RUL and LUR.
T(3,3) = 3: the 3 walks from (0,0) to (2,1) of three steps are RRU, RUR and URR.
.
.
*......*......*......y......*......*......*
.
.
*......3......*......8......*......3......*
.
.
*......*......*......o......*......*......* x-axis
.
.

F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO],2018; Table 2.1 for k=1.
Colin Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018; Graphs Combin., 33 (2017), 103-122.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Serge Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, J. Combin. Theory Ser. A, 77(2): 193-221, 1997.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Toufik Mansour and Yidong Sun, Identities involving Narayana polynomials and Catalan numbers, arXiv:0805.1274 [math.CO], 2008.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A002057 (row sums), A001263, A108838, A145597, A145598, A145599, A145600.

/* As triangle */  [[2/(n+1)*Binomial(n+1,k+1)*Binomial(n+1,k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Sep 15 2018

T:= (n, k) -> 2/(n+1)*binomial(n+1, k+1)*binomial(n+1, k-1):
for n from 1 to 10 do seq(T(n,k), k = 1..n) end do;

t[n_, k_]:=2/(n+1) Binomial[n+1, k+1] Binomial[n+1, k-1]; Table[t[n, k], {n, 3, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Sep 15 2018 *)

T(n,k) = 2/(n + 1)*binomial(n + 1,k + 1)*binomial(n + 1,k - 1) for 1 <= k <= n. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n - 1,1).
O.g.f. for column (k + 2): 2/(k + 1) * y^(k+2)/(1 - y)^(k+4) * Jacobi_P(k,2,1,(1 + y)/(1 - y)). The column generating functions begin: column 2: 2*y^2/(1 - y)^4; column 3: y^3*(3 + 2*y)/(1 - y)^6; column 4: y^4*(4 + 8*y + 2*y^2)/(1 - y)^8; the polynomials in the numerators are the row generating polynomials of array A108838.
O.g.f. for array: 1/(2*x*y^3) * (((1 + x)*y - 1)*sqrt(1 - 2*(1 + x)*y + (y - x*y)^2) + x^2*y^2 - 2*x*y + (1 - y)^2) = x*y + (2*x + 2*x^2)*y^2 + (3*x + 8*x^2 + 3*x^3)*y^3 + (4*x + 20*x^2 + 20*x^3 + 4*x^4)*y^4 + ... .
Row sums A002057.
Identities for row polynomials R_n(x) = Sum_{k = 1..n} T(n,k)*x^k (compare with the results in section 1 of [Mansour & Sun]):
x*R_(n-1)(x) = 2*(n - 1)/((n + 1)*(n + 2)) * Sum_{k = 0..n} binomial(n + 2,k) * binomial(2*n - k,n) * (x - 1)^k;
R_n(x) = Sum_{k = 0..floor((n-1)/2)} binomial(n, 2*k + 1) * Catalan(k + 1) * x^(k+1)*(1 + x)^(n-2k-1);
Sum_{k = 1..n} (-1)^(n-k)*binomial(n,k)*R_k(x)*(1 + x)^(n-k) = x^m*Catalan(m) if n = 2*m - 1 is odd, otherwise the sum is zero.
Sum_{k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n+1) = 4/(n + 3)*binomial(2*n + 1,n - 1)*x^(n+1) = A002057(n-1)*x^(n+1).
Row generating polynomial R_(n+1)(x) = 2/(n + 2)*x*(1 - x)^n * Jacobi_P(n,2,2,(1 + x)/(1 - x)). - Peter Bala, Oct 31 2008
G.f. satisfies x^3*y*A(x,y)^2-A(x,y)*(x^2*y^2+(-2)*x*y+x^2+(-2)*x+1)+x = 0. - Vladimir Kruchinin, Oct 11 2020
The array can be extended to negative values of n: T(-n,k) = 2*binomial(-n + 1, k + 1)*binomial(-n + 1, k - 1)/(-n + 1) = -A108838(n+k-1,k-1) for n >= 2. - Peter Bala, Apr 26 2022
From Sergii Voloshyn, Dec 18 2024: (Start)
Let E be the operator x^2*D*(1/x)*D, where D denotes the derivative operator d/dx. Then 48(1+n)/((n+2)!(n+4)!)* E^n(x^2/(1 - x)^5) = (row n generating polynomial)/(1 - x)^(2*n+5) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
For example, when n = 3 we have 1/3150*E^3(x^2/(1 - x)^5) = x^2 (4 + 20x + 20x^2 + 4x^3)/(1 - x)^11. (End)

A145601 a(n) is the number of walks from (0,0) to (0,2) that remain in the upper half-plane y >= 0 using 2*n unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 15, 189, 2352, 29700, 382239, 5010005, 66745536, 901995588, 12342120700, 170724392916, 2384209771200, 33577620944400, 476432168185575, 6805332732133125, 97790670976838400, 1412830549632694500

Offset: 1

Peter Bala, Oct 15 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145602 and A145603. This sequence is the central column taken from triangle A145597, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 2.

			a(2) = 15: the 15 walks from (0,0) to (0,2) of four steps are:
UUUD, UULR, UURL, UUDU, URUL, ULUR, URLU, ULRU,RUUL, LUUR,
RLUU, LRUU, RULU, LURU and UDUU.

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

A000891, A145597, A145600, A145602, A145603.

with(combinat):
a(n) = 3/(2*n+1)*binomial(2*n+1,n+2)*binomial(2*n+1,n-1);
seq(a(n),n = 1..19);

a(n) = 3/(2*n+1)*binomial(2*n+1,n+2)*binomial(2*n+1,n-1).

A145602 a(n) is the number of walks from (0,0) to (0,3) that remain in the upper half-plane y >= 0 using 2*n +1 unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 24, 392, 5760, 81675, 1145144, 16032016, 225059328, 3173688180, 44986664800, 641087516256, 9183622822400, 132211882468575, 1912322889603000, 27781440618420000, 405248874740582400, 5933888308457316900

Offset: 1

Peter Bala, Oct 15 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145601 and A145603. This sequence is the central column taken from the triangle A145598, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 3.

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

A000891, A145598, A145600, A145601, A145603.

with(combinat):
a(n) = 2/(n+1)*binomial(2*n+2,n+3)*binomial(2*n+2,n-1);
seq(a(n),n = 1..19);

a(n) = 2/(n+1)*binomial(2*n+2,n+3)*binomial(2*n+2,n-1).

A145603 a(n) is the number of walks from (0,0) to (0,4) that remain in the upper half-plane y >= 0 using 2*n +2 unit steps either up (U), down (D), left (L) or right (R).

Original entry on oeis.org

1, 35, 720, 12375, 196625, 3006003, 45048640, 668144880, 9859090500, 145173803500, 2136958387520, 31479019635375, 464342770607625, 6861343701121875, 101583106970400000, 1507019252941540800

Offset: 1

Peter Bala, Oct 15 2008

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145600, A145601 and A145602. This sequence is the central column taken from the triangle A145599, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 4.

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

A000891, A145599, A145600, A145601, A145602.

with(combinat):
a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1);
seq(a(n),n = 1..19);

a(n) = 5/(2*n+3)*binomial(2*n+3,n+4)*binomial(2*n+3,n-1).

G.f.: x*hypergeom([5/2, 3, 3, 7/2], [1, 6, 6], 16*x). - Stefano Spezia, Aug 26 2025

A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 20, 0, 15, 0, 1, 0, 75, 0, 24, 0, 1, 175, 0, 189, 0, 35, 0, 1, 0, 784, 0, 392, 0, 48, 0, 1, 1764, 0, 2352, 0, 720, 0, 63, 0, 1, 0, 8820, 0, 5760, 0, 1215, 0, 80, 0, 1, 19404, 0, 29700, 0, 12375, 0, 1925, 0, 99, 0, 1

Offset: 0

Peter Luschny, Dec 07 2024

Consider square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. T(n, k) is the number of walks with length n and height k. The number of walks with positive height is A378060, and with nonnegative height is A018224. Walks of odd length can never have an even height, and walks of even length cannot have an odd height. The Python program below generates the walks.

			Triangle starts:
  0  [   1]
  1  [   0,    1]
  2  [   3,    0,    1]
  3  [   0,    8,    0,    1]
  4  [  20,    0,   15,    0,   1]
  5  [   0,   75,    0,   24,   0,    1]
  6  [ 175,    0,  189,    0,  35,    0,  1]
  7  [   0,  784,    0,  392,   0,   48,  0,  1]
  8  [1764,    0, 2352,    0, 720,    0, 63,  0, 1]
  9  [   0, 8820,    0, 5760,   0, 1215,  0, 80, 0, 1]
.
The 15 walks with length 4 and height 2 are: 'NNNS', 'NNSN', 'NNWE', 'NNEW', 'NSNN', 'NWNE', 'NWEN', 'NENW', 'NEWN', 'WNNE', 'WNEN', 'WENN', 'ENNW', 'ENWN', 'EWNN'.

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

The columns are aerated rows of A378062. See also: A000891, A145600, A145601, A145602, A145603.

Cf. A018224 (row sums), A378060.

T := (n, k) -> ifelse((n - k)::odd, 0, binomial(n+1, (n-k)/2)^2*(k+1)/(n+1)):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

T[n_, k_] := If[EvenQ[n-k],Binomial[n + 1, (n - k)/2]^2*(k + 1)/(n + 1), 0]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

# Creates the table by counting the heights of square lattice walks. For illustration only.
from dataclasses import dataclass
@dataclass
class Z: w: str = ""; r: int = 0; i: int = 0
def Trow(n: int) -> list[int]:
    W = [Z()]
    row = [0] * (n + 1)
    for x in W:
        if len(x.w) == n:
            if x.r == 0: row[x.i] += 1
        else:
            for s in "NSWE":
                r = i = 0
                match s:
                    case "W": r = 1
                    case "E": r = -1
                    case "N": i = 1
                    case "S": i = -1
                if x.i + i >= 0:
                    W.append(Z(x.w + s, x.r + r, x.i + i))
    return row
for n in range(10): print(f"[{n}] {Trow(n)}")

A214457 Table read by antidiagonals in which entry T(n,k) in row n and column k gives the number of possible rhombus tilings of an octagon with interior angles of 135 degrees and sequences of side lengths {n, k, 1, 1, n, k, 1, 1} (as the octagon is traversed), n,k in {1,2,3,...}.

Original entry on oeis.org

8, 20, 20, 40, 75, 40, 70, 210, 210, 70, 112, 490, 784, 490, 112, 168, 1008, 2352, 2352, 1008, 168, 240, 1890, 6048, 8820, 6048, 1890, 240, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 440, 5445, 29040, 76230, 104544, 76230, 29040, 5445, 440

Offset: 1

L. Edson Jeffery, Jul 18 2012

Proof of the formula for T(n,k) is given in [Elnitsky].

So-called "generalized Narayana numbers" (see A145596), linking rhombus tilings of polygons to certain walks or paths through the square lattice.

			See [Jeffery]. T(1,1) = 8 because there are eight ways to tile the proposed octagon with rhombuses.
Table begins as
    8    20    40     70    112  ...
   20    75   210    490   1008  ...
   40   210   784   2352   6048  ...
   70   490  2352   8820  27720  ...
  112  1008  6048  27720  76230  ...
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Serge Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups (preprint), J. Combin. Theory Ser. A, Vol. 77, Issue 2, 193-221 (1997).
L. E. Jeffery, Worked out example for A214457(1,1)=8
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Empirical: T(1,n) = T(n,1) = 2*A000292(n+1); T(2,n) = T(n,2) = A006411(n+1); T(n,n) = A145600(n+1).

Table[2*(# + k + 1)!*(# + k + 2)!/(#!*k!*(# + 2)!*(k + 2)!) &[n - k + 1], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Feb 26 2024 *)

T(n,k) = 2*(n+k+1)!*(n+k+2)!/[n!*k!*(n+2)!*(k+2)!].

A227169 a(n) = 3((2n+2)!)^2 / (n!(n+1)!(n+2)!*(n+3)!).

Original entry on oeis.org

1, 6, 45, 392, 3780, 39204, 429429, 4907760, 58023108, 705264040, 8772399636, 111263122656, 1434941066000, 18775651948200, 248797110637125, 3333772874210400, 45115597383228900, 615974564891763000, 8477309210264363700, 117511846058893572000

Offset: 0

Karol A. Penson, Jul 12 2013

nonn

T. D. Noe, Table of n, a(n) for n = 0..200

seq(3*((2*n+2)!)^2/(n!*(n+1)!*(n+2)!*(n+3)!),n=0..15);

Table[3*((2*n + 2)!)^2/(n!*(n + 1)!*(n + 2)!*(n + 3)!), {n, 0, 20}] (* T. D. Noe, Jul 12 2013 *)

def a(n): return 3*(n+1)*(n+2)^2*(n+3)^3*gamma(2*n+3)^2/gamma(n+4)^4
[a(n) for n in (0..16)]  # Peter Luschny, Jul 12 2013

In Maple notation,
  ogf(z) = 3/(4*z^2) +(1/12288)*(-98304*z^2-2048*z+512)*EllipticK(4*sqrt(z))/(z^3*Pi) +(1/12288)*(-20480*z-512)*EllipticE(4*sqrt(z))/(z^3*Pi);
egf(z)=hypergeom([3/2, 3/2, 2], [1, 3, 4], 16*z), a 3F3 hypergeometric function.
Integral representation as the n-th moment of a signed function w(x) of bounded variation, 0<=x<=16: a(n) = Integral_{x=0..16}x^n*w(x), n>=0, where w(x) is the Meijer G function, w(x) = -3*MeijerG([[0], [2, 3]], [[1/2, 1/2], [1]], (1/16)*x)/Pi, satisfying w(16)=w(0)=0, w(x)<0 for x < 0.47.
The above Meijer G function cannot be represented by any other special function.
(n+3)*(n+2)*a(n) -18*(n+1)^2*a(n-1) +8*(2*n-1)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013
a(n) = 3*A135389(n)/((n+2)*(n+3)) = 3*A145600(n+1)/(n+3). - R. J. Mathar, Jul 14 2013

A268085 a(n) = Catalan(n)^2*n.

Original entry on oeis.org

0, 1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700, 4106497099278420000, 59341164471850545900, 861753537765219528000

Offset: 0

Ralf Steiner, Jan 26 2016

The series whose terms are the quotients a(n)/A013709(n) is convergent to 1-3/Pi.(see formula).
Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 1-3/Pi. Q.E.D.
Apart from inclusion of a(0) the same as A145600. - R. J. Mathar, Feb 07 2016

			For n=3 the a(3)= 75.

Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).

Cf. A000108, A013709.

[Catalan(n)^2*n: n in [0..20]]; // Vincenzo Librandi, Jan 26 2016

Mathematica
```
Table[CatalanNumber[n]^2 n, {n, 0, 20}]
```

a(n) = n*(binomial(2*n, n)/(n+1))^2; \\ Altug Alkan, Jan 26 2016

Sum_{n>=0} a(n)/A013709(n) = 1 - 3/Pi (see A089491).

Corrected and extended by Vincenzo Librandi, Jan 26 2016

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 8, 20, 0, 1, 15, 75, 175, 0, 1, 24, 189, 784, 1764, 0, 1, 35, 392, 2352, 8820, 19404, 0, 1, 48, 720, 5760, 29700, 104544, 226512, 0, 1, 63, 1215, 12375, 81675, 382239, 1288287, 2760615, 0, 1, 80, 1925, 24200, 196625, 1145144, 5010005, 16359200, 34763300

Offset: 0

Peter Luschny, Dec 07 2024

			Array A(n, k) starts:
  [0] 0, 1,  3,   20,   175,    1764,    19404, ... A000891
  [1] 0, 1,  8,   75,   784,    8820,   104544, ... A145600
  [2] 0, 1, 15,  189,  2352,   29700,   382239, ... A145601
  [3] 0, 1, 24,  392,  5760,   81675,  1145144, ... A145602
  [4] 0, 1, 35,  720, 12375,  196625,  3006003, ... A145603
  [5] 0, 1, 48, 1215, 24200,  429429,  7154784, ...
  [6] 0, 1, 63, 1925, 44044,  869505, 15767024, ...
  [7] 0, 1, 80, 2904, 75712, 1656200, 32626944, ...
.
Seen as a triangle, T(n, k) = A(n-k, k). Compare the descending antidiagonals of A378061.
  [0] 0;
  [1] 0, 1;
  [2] 0, 1,  3;
  [3] 0, 1,  8,  20;
  [4] 0, 1, 15,  75,  175;
  [5] 0, 1, 24, 189,  784,  1764;
  [6] 0, 1, 35, 392, 2352,  8820,  19404;
  [7] 0, 1, 48, 720, 5760, 29700, 104544, 226512;

Rows: A000891, A145600, A145601, A145602, A145603.

Columns: A005563, A005565, A378061.

A := (n, k) -> ifelse(k = 0, 0, (n + 1)*binomial(2*k + n - 1, k - 1)^2/(2*k + n - 1)):
for n from 0 to 7 do seq(A(n, k), k = 0..7);

A[n_, k_] := If[k==0, 0, (n + 1)*Binomial[2*k + n - 1, k - 1]^2 / (2*k + n - 1)]; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

cp's OEIS Frontend

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

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A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).

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A145601 a(n) is the number of walks from (0,0) to (0,2) that remain in the upper half-plane y >= 0 using 2*n unit steps either up (U), down (D), left (L) or right (R).

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A145602 a(n) is the number of walks from (0,0) to (0,3) that remain in the upper half-plane y >= 0 using 2*n +1 unit steps either up (U), down (D), left (L) or right (R).

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A145603 a(n) is the number of walks from (0,0) to (0,4) that remain in the upper half-plane y >= 0 using 2*n +2 unit steps either up (U), down (D), left (L) or right (R).

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A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

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A214457 Table read by antidiagonals in which entry T(n,k) in row n and column k gives the number of possible rhombus tilings of an octagon with interior angles of 135 degrees and sequences of side lengths {n, k, 1, 1, n, k, 1, 1} (as the octagon is traversed), n,k in {1,2,3,...}.

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

A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2binomial(n + 1, k + 1)binomial(n + 1, k - 1)/(n + 1).

A227169 a(n) = 3((2n+2)!)^2 / (n!(n+1)!(n+2)!*(n+3)!).

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.