OFFSET
0,9
COMMENTS
Also related to dissections of polygons and enumeration of trees.
Number of dissections of a polygon into n (k+2)-gons by nonintersecting diagonals. All dissections are counted separately. See A295260 for nonequivalent solutions up to rotation and reflection. - Andrew Howroyd, Nov 20 2017
Number of rooted polyominoes composed of n (k+2)-gonal cells of the hyperbolic (Euclidean for k=0) regular tiling with Schläfli symbol {k+2,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. For k>0, a stereographic projection of the {k+2,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Peter Hilton and Jean Pedersen, Catalan Numbers, Their Generalization, and Their Uses, The Mathematical Intelligencer, March 1991, Volume 13, Issue 2, pp. 64-75.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
FORMULA
T(n, k) = binomial(n*(k+1), n)/(n*k+1) = A071201(n, k*n) = A071201(n, k*n+1) = A071202(n, k*n+1) = A062993(n+k-1, k-1).
If P(k,x) = Sum_{n>=0} T(n,k)*x^n is the g.f. of column k (k>=0), then P(k,x) = exp(1/(k+1)*(Sum_{j>0} (1/j)*binomial((k+1)*j,j)*x^j)). - Werner Schulte, Oct 13 2015
EXAMPLE
Rows start:
===========================================================
n\k| 0 1 2 3 4 5 6
---|-------------------------------------------------------
0 | 1, 1, 1, 1, 1, 1, 1 ...
1 | 1, 1, 1, 1, 1, 1, 1 ...
2 | 1, 2, 3, 4, 5, 6, 7 ...
3 | 1, 5, 12, 22, 35, 51, 70 ...
4 | 1, 14, 55, 140, 285, 506, 819 ...
5 | 1, 42, 273, 969, 2530, 5481, 10472 ...
6 | 1, 132, 1428, 7084, 23751, 62832, 141778 ...
7 | 1, 429, 7752, 53820, 231880, 749398, 1997688 ...
8 | 1, 1430, 43263, 420732, 2330445, 9203634, 28989675 ...
...
MAPLE
A:= (n, k)-> binomial((k+1)*n, n)/(k*n+1):
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 25 2015
MATHEMATICA
T[n_, k_] = Binomial[n(k+1), n]/(k*n+1); Flatten[Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Apr 08 2016 *)
PROG
(PARI) T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ Andrew Howroyd, Nov 20 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, May 20 2002
STATUS
approved