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A361745
Square array of circular Delannoy numbers A(i,j) (i >= 0, j >= 0) read by antidiagonals.
2
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
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.
Sequence in context: A161126 A128562 A034368 * A296157 A113582 A347147
KEYWORD
nonn,tabl
AUTHOR
Noah Snyder, Mar 22 2023
STATUS
approved