|
|
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.
|
|
19
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
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):
|
|
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
|
Reflected version of A062993 (which is the main entry).
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|