A361745 - OEIS

%I #25 Mar 24 2023 02:55:26

%S 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,

%T 12,100,264,264,100,12,1,1,14,144,510,768,510,144,14,1,1,16,196,876,

%U 1800,1800,876,196,16,1,1,18,256,1386,3648,5010,3648,1386,256,18,1

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

%C 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.

%C 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.

%H Nate Harman, Andrew Snowden, and Noah Snyder, <a href="https://arxiv.org/abs/2303.10814">The circular Delannoy Category</a>, arxiv: 2303.10814 [math.RT], 2023.

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

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

%F 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.

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

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

%F A(n,m) = 2*n*m*hypergeom([1-n, 1-m], [2], 2) for n,m >= 1. - _Peter Luschny_, Mar 23 2023

%e The square array A(n,m) (n >= 0, m >= 0) begins:

%e   1, 1,  1,   1,   1,    1,    1,    1,     1,     1, ...

%e   1, 2,  4,   6,   8,   10,   12,   14,    16,    18, ...

%e   1, 4, 16,  36,  64,  100,  144,  196,   256,   324, ...

%e   1, 6, 36, 114, 264,  510,  876, 1386,  2064,  2934, ...

%e   1, 8, 64, 264, 768, 1800, 3648, 6664, 11264, 17928, ...

%e .

%e The triangle T(n,m) (0 <= m <= n) begins:

%e   [0] 1;

%e   [1] 1,  1;

%e   [2] 1,  2,   1;

%e   [3] 1,  4,   4,   1;

%e   [4] 1,  6,  16,   6,    1;

%e   [5] 1,  8,  36,  36,    8,    1;

%e   [6] 1, 10,  64, 114,   64,   10,   1;

%e   [7] 1, 12, 100, 264,  264,  100,  12,   1;

%e   [8] 1, 14, 144, 510,  768,  510, 144,  14,  1;

%e   [9] 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1;

%p A := (n, k) -> `if`(n*k=0, 1, 2*n*k*hypergeom([1 - n, 1 - k], [2], 2)):

%p seq(print(seq(simplify(A(n, k)), k = 0..9)), n=0..4); # _Peter Luschny_, Mar 23 2023

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

%o (Python)

%o from math import comb

%o def A361745_A(n,m): # compute square array A(n,m)

%o     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

%Y Circular analog of A008288.

%Y Main diagonal: A361743.

%Y Row sums: A361758.

%Y Cf. A142978, A104698.

%K nonn,tabl

%O 0,5

%A _Noah Snyder_, Mar 22 2023