This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361745 #25 Mar 24 2023 02:55:26
%S A361745 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 A361745 12,100,264,264,100,12,1,1,14,144,510,768,510,144,14,1,1,16,196,876,
%U A361745 1800,1800,876,196,16,1,1,18,256,1386,3648,5010,3648,1386,256,18,1
%N A361745 Square array of circular Delannoy numbers A(i,j) (i >= 0, j >= 0) read by antidiagonals.
%C A361745 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 A361745 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 A361745 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 A361745 A(n,m) = A(m,n).
%F A361745 A(n,m) = Sum_{k=0..min(n,m)} binomial(n,k)*binomial(m,k)*k*2^k for n >= 1.
%F A361745 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 A361745 G.f.: 2*x*y/(1-x-y-x*y)^2 (valid for n,m > 1).
%F A361745 For n,m >= 1, A(n,m) = 2*n*A142978(n,m).
%F A361745 A(n,m) = 2*n*m*hypergeom([1-n, 1-m], [2], 2) for n,m >= 1. - _Peter Luschny_, Mar 23 2023
%e A361745 The square array A(n,m) (n >= 0, m >= 0) begins:
%e A361745 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A361745 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
%e A361745 1, 4, 16, 36, 64, 100, 144, 196, 256, 324, ...
%e A361745 1, 6, 36, 114, 264, 510, 876, 1386, 2064, 2934, ...
%e A361745 1, 8, 64, 264, 768, 1800, 3648, 6664, 11264, 17928, ...
%e A361745 .
%e A361745 The triangle T(n,m) (0 <= m <= n) begins:
%e A361745 [0] 1;
%e A361745 [1] 1, 1;
%e A361745 [2] 1, 2, 1;
%e A361745 [3] 1, 4, 4, 1;
%e A361745 [4] 1, 6, 16, 6, 1;
%e A361745 [5] 1, 8, 36, 36, 8, 1;
%e A361745 [6] 1, 10, 64, 114, 64, 10, 1;
%e A361745 [7] 1, 12, 100, 264, 264, 100, 12, 1;
%e A361745 [8] 1, 14, 144, 510, 768, 510, 144, 14, 1;
%e A361745 [9] 1, 16, 196, 876, 1800, 1800, 876, 196, 16, 1;
%p A361745 A := (n, k) -> `if`(n*k=0, 1, 2*n*k*hypergeom([1 - n, 1 - k], [2], 2)):
%p A361745 seq(print(seq(simplify(A(n, k)), k = 0..9)), n=0..4); # _Peter Luschny_, Mar 23 2023
%t A361745 a[n_Integer?Positive, m_Integer?Positive] := Sum[k Binomial[n, k] Binomial[m, k] 2^k, {k, 1, Min[n,m]}]
%o A361745 (Python)
%o A361745 from math import comb
%o A361745 def A361745_A(n,m): # compute square array A(n,m)
%o A361745 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 A361745 Circular analog of A008288.
%Y A361745 Main diagonal: A361743.
%Y A361745 Row sums: A361758.
%Y A361745 Cf. A142978, A104698.
%K A361745 nonn,tabl
%O A361745 0,5
%A A361745 _Noah Snyder_, Mar 22 2023