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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. 12
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

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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-Fran├žois 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.

Columns include A000012, A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A062744.

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

Cf. A295260.

Sequence in context: A299045 A124530 A243631 * A305962 A144150 A124560

Adjacent sequences:  A070911 A070912 A070913 * A070915 A070916 A070917

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, May 20 2002

STATUS

approved

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Last modified November 17 06:06 EST 2019. Contains 329217 sequences. (Running on oeis4.)