A361745 Square array of circular Delannoy numbers A(i,j) (i >= 0, j >= 0) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 16, 6, 1, 1, 8, 36, 36, 8, 1, 1, 10, 64, 114, 64, 10, 1, 1, 12, 100, 264, 264, 100, 12, 1, 1, 14, 144, 510, 768, 510, 144, 14, 1, 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1, 1, 18, 256, 1386, 3648, 5010, 3648, 1386, 256, 18, 1

Offset: 0

Author

Noah Snyder, Mar 22 2023

Keywords

• nonn
• tabl

Comments

An (n,m) Delannoy loop is an oriented unbased loop on a toroidal grid with points labeled by Z/n x Z/m composed of steps of the form (1,0), (0,1), and (1,1), and which loops around the torus exactly once in each of the x-direction and the y-direction. The circular Delannoy numbers count the number of (n,m) Delannoy loops. This array is a modification of the ordinary Delannoy numbers A008288.

Dimensions of hom spaces Hom(S^{{i}}, S^{{j}}) in the circular Delannoy category attached to the oligomorphic group of order preserving self-bijections of the circle.

Examples

			The square array A(n,m) (n >= 0, m >= 0) begins:
  1, 1,  1,   1,   1,    1,    1,    1,     1,     1, ...
  1, 2,  4,   6,   8,   10,   12,   14,    16,    18, ...
  1, 4, 16,  36,  64,  100,  144,  196,   256,   324, ...
  1, 6, 36, 114, 264,  510,  876, 1386,  2064,  2934, ...
  1, 8, 64, 264, 768, 1800, 3648, 6664, 11264, 17928, ...
.
The triangle T(n,m) (0 <= m <= n) begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  2,   1;
  [3] 1,  4,   4,   1;
  [4] 1,  6,  16,   6,    1;
  [5] 1,  8,  36,  36,    8,    1;
  [6] 1, 10,  64, 114,   64,   10,   1;
  [7] 1, 12, 100, 264,  264,  100,  12,   1;
  [8] 1, 14, 144, 510,  768,  510, 144,  14,  1;
  [9] 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1;
		

Links

• Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Crossrefs

Circular analog of A008288.

Main diagonal: A361743.

Row sums: A361758.

Cf. A142978, A104698.

Programs

Maple

A := (n, k) -> `if`(n*k=0, 1, 2*n*k*hypergeom([1 - n, 1 - k], [2], 2)):
seq(print(seq(simplify(A(n, k)), k = 0..9)), n=0..4); # Peter Luschny, Mar 23 2023

Mathematica

a[n_Integer?Positive, m_Integer?Positive] := Sum[k Binomial[n, k] Binomial[m, k] 2^k, {k, 1, Min[n,m]}]

Python

from math import comb
def A361745_A(n,m): # compute square array A(n,m)
    return 1 if not(m and n) else sum(comb(n-1,i)*comb(m+i,n) for i in range(max(n-m,0),n))*n<<1 # Chai Wah Wu, Mar 23 2023

Formula

A(n,m) = A(m,n).

A(n,m) = Sum_{k=0..min(n,m)} binomial(n,k)*binomial(m,k)*k*2^k for n >= 1.

A(n,m) = n*(D(n,m-1) + D(n-1,m-1)) = n*(D(n,m) - D(n-1,m)) for n,m >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.

G.f.: 2*x*y/(1-x-y-x*y)^2 (valid for n,m > 1).

For n,m >= 1, A(n,m) = 2*n*A142978(n,m).

A(n,m) = 2*n*m*hypergeom([1-n, 1-m], [2], 2) for n,m >= 1. - Peter Luschny, Mar 23 2023