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User: Noah Snyder

Noah Snyder has authored 2 sequences.

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

Noah Snyder, Mar 22 2023

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.

			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;

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

Circular analog of A008288.
Main diagonal: A361743.
Row sums: A361758.
Cf. A142978, A104698.

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

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

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

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

A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

Original entry on oeis.org

1, 2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, 1779856128, 10799942322, 65336473104, 394246725570, 2373580947456, 14262064668738, 85546366040592, 512323096241714, 3063932437123840, 18300660294266322, 109183694129335056

Offset: 0

Noah Snyder, Mar 22 2023

nonn

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 central circular Delannoy numbers count the number of (n,n) Delannoy loops. This is a modification of the ordinary central Delannoy numbers A001850.

Dimensions of endomorphism algebras End(S^{{n}}) in the circular Delannoy category attached to the oligomorphic group of order-preserving self-bijections of the circle.

			When n=2 see Figure 3 of "The circular Delannoy Category".

Winston de Greef, Table of n, a(n) for n = 0..1296
Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Circular analog of A001850.
Main diagonal of A361745.
Cf. A008288, A108666, A002003, A047781.

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

a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ Winston de Greef, Mar 22 2023

from math import comb
def A361743(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = Sum_{k=0..n} binomial(n,k)^2*k*2^k for n >= 1.
a(n) = n*(D(n,n-1) + D(n-1,n-1)) = n*(D(n,n) - D(n-1,n)) for n >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.
a(n) = 2*A108666(n) for n >= 1.
From Alois P. Heinz, Mar 22 2023: (Start)
G.f.: 1 + 2*(1-x)*x/sqrt(x^2-6*x+1)^3.
a(n) = n*A002003(n) for n >= 1.
a(n) = 2*n*A047781(n) for n >= 1. (End)
a(n) = 2*n^2*hypergeom([1 - n, 1 - n], [2], 1) for n >= 1. - Peter Luschny, Mar 22 2023

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User: Noah Snyder

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

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