A071202 -id:A071202 - cp's OEIS frontend

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

Henry Bottomley, May 20 2002

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

			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 ...
...

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.

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).

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

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 *)

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

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

A071201 Array A(n,k) read by antidiagonals giving number of paths up and right from (0,0) to (n,k) where x/y<=n/k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 14, 7, 4, 1, 1, 4, 12, 14, 14, 12, 4, 1, 1, 5, 12, 23, 42, 23, 12, 5, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 22, 55, 66, 132, 66, 55, 22, 6, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1, 7, 26, 76, 143, 227, 429, 227, 143, 76, 26, 7, 1

Offset: 1

Henry Bottomley, May 16 2002

			Table starts:
1, 1, 1,  1,  1,  1, ...
1, 2, 2,  3,  3,  4, ...
1, 2, 5,  5,  7, 12, ...
1, 3, 5, 14, 14, 23, ...
1, 3, 7, 14, 42, 42, ...
...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Jean-Christophe Aval, François Bergeron, Interlaced rectangular parking functions, arXiv:1503.03991 [math.CO], 2015.

Cf. A260419, A298072.

b:= proc(x, y, r) option remember; `if`(y<0 or y>x*r, 0,
      `if`(x=0, 1, b(x-1, y, r) +b(x, y-1, r)))
    end:
A:= (n, k)-> `if`(kAlois P. Heinz, Mar 20 2015

b[x_, y_, r_] := b[x, y, r] = If[y < 0 || y > x*r, 0, If[x == 0, 1, b[x - 1, y, r] + b[x, y - 1, r]]]; A[n_, k_] := If[k < n, b[k, n, n/k], b[n, k, k/n]]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Jan 30 2016, after Alois P. Heinz *)

Some identities: A(n,k) = A(k,n); A(n,m*n) = A(n,m*n+1); A(n,n) = A000108(n); if n and k are coprime then A(n,k) = A071202(n,k).

Sum_{k=1..n-1} A(n-k,k) = A298072(n)-2 for n>0. - Lee A. Newberg, Jan 18 2018

A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).

Original entry on oeis.org

1, 1, 2, 12, 140, 2530, 62832, 1997688, 77652024, 3573805950, 190223180840, 11502251937176, 779092434772236, 58448142042957576, 4811642166029230560, 431306008583779517040, 41820546066482630185200

Offset: 0

Paul Barry, Dec 22 2003

Diagonal of array T(n,k) = binomial(kn,n)/(1+(k-1)n).
Number of paths up and left from (0,0) to (n^2-n,n) where x/y <= n-1 for all intermediate points. - Henry Bottomley, Dec 25 2003
Empirical: In the ring of symmetric functions over the fraction field Q(q, t), letting s(1^n) denote the Schur function indexed by (1^n), a(n) is equal to the coefficient of s(n) in nabla^(n)s(1^n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions, and s(n) denotes the Schur function indexed by the integer partition (n) of n. - John M. Campbell, Apr 06 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A002061, A014062, A062993, A070914, A071201, A071202.

List([0..20],n->Binomial(n^2,n)/(1+(n-1)*n)); # Muniru A Asiru, Apr 08 2018

[Binomial(n^2, n)/(1+(n-1)*n): n in [0..20]]; // Vincenzo Librandi, Apr 07 2018

A091144 := proc(n)
    binomial(n^2,n)/(1+n*(n-1)) ;
end proc: # R. J. Mathar, Feb 14 2015

Table[Binomial[n^2, n] / (n (n - 1) + 1), {n, 0, 20}] (* Vincenzo Librandi, Apr 07 2018 *)

a(n) = binomial(n^2, n)/(n*(n-1)+1); \\ Altug Alkan, Apr 06 2018

From Henry Bottomley, Dec 25 2003: (Start)
a(n) = A014062(n)/A002061(n);
a(n) = A062993(n-2, n);
a(n) = A070914(n, n-1);
a(n) = A071201(n, n^2-n);
a(n) = A071201(n, n^2-n+1);
a(n) = A071202(n, n^2-n+1). (End)

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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.

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A071201 Array A(n,k) read by antidiagonals giving number of paths up and right from (0,0) to (n,k) where x/y<=n/k.

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Maple

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Formula

A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).

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