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,...}.

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

Author

L. Edson Jeffery, Jul 18 2012

Keywords

• nonn
• tabl

Comments

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.

Examples

			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  ...
  ...
		

Links

• 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.

Crossrefs

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).

Programs

Mathematica

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

Formula

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