A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 14, 1, 1, 1, 5, 22, 55, 42, 1, 1, 1, 6, 35, 140, 273, 132, 1, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 1, 1, 9, 92, 819, 5481, 23751, 53820, 43263, 4862, 1, 1, 1, 10, 117, 1240

Offset: 0

Author

Henry Bottomley, May 20 2002

Keywords

• nonn
• tabl

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

Examples

			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 ...
...
		

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.

Crossrefs

Rows include A000012 (twice), A000027, A000326.

Columns include A000012, A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968.

Reflected version of A062993 (which is the main entry).

Cf. A295260.

Polyominoes: A295224 (oriented), A295260 (unoriented).

Programs

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 *)

PARI

T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ Andrew Howroyd, Nov 20 2017

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