

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,...}.


1



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
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OFFSET

1,1


COMMENTS

Proof of the formula for T(n,k) is given in [Elnitsky].
Socalled "generalized Narayana numbers" (see A145596), linking rhombus tilings of polygons to certain walks or paths through the square lattice.
(Start) 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...;
etc. (End)


LINKS

Table of n, a(n) for n=1..45.
Serge Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups (preprint), J. Combin. Theory Ser. A, Vol. 77, Issue 2, 193221 (1997).
L. E. Jeffery, Worked out example for A214457(1,1)=8


FORMULA

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


EXAMPLE

See [Jeffery]. T(1,1) = 8 because there are eight ways to tile the proposed octagon with rhombuses.


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).
Sequence in context: A115147 A302241 A022700 * A205226 A205318 A100212
Adjacent sequences: A214454 A214455 A214456 * A214458 A214459 A214460


KEYWORD

nonn,tabl


AUTHOR

L. Edson Jeffery, Jul 18 2012


STATUS

approved



