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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
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
Rows include A000012 (twice), A000027, A000326.
Reflected version of A062993 (which is the main entry).
Cf. A295260.
Polyominoes: A295224 (oriented), A295260 (unoriented).
Sequence in context: A299045 A124530 A243631 * A305962 A144150 A124560
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, May 20 2002
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)