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

0,5

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.

Links

Table of n, a(n) for n=0..65.

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

Example

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;

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]}]

Prog

(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

Crossrefs

Circular analog of A008288.

Main diagonal: A361743.

Row sums: A361758.

Cf. A142978, A104698.

Sequence in context: A161126 A128562 A034368 * A296157 A113582 A347147

Adjacent sequences: A361742 A361743 A361744 * A361746 A361747 A361748

Keyword

nonn,tabl

Author

Noah Snyder, Mar 22 2023

Status

approved