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 A143670 #20 Jan 28 2022 07:44:02
%S A143670 1,1,1,1,2,1,1,7,3,1,1,42,26,4,1,1,429,628,70,5,1,1,7436,41784,5102,
%T A143670 155,6,1,1,218348,7517457,1507128,28005
%N A143670 Array of higher spin alternating sign matrices, read by antidiagonals.
%C A143670 Adapted from Table 1, p. 5: |ASM(n, r)|, where A[k,n] = |ASM(n, k)|. Abstract: We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions.
%C A143670 The case r = 1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.
%H A143670 Roger E. Behrend and Vincent A. Knight, <a href="http://arxiv.org/abs/0708.2522">Higher Spin Alternating Sign Matrices</a>, arXiv:0708.2522 [math.CO], 2007.
%H A143670 Roger E. Behrend, Vincent A. Knight, <a href="https://doi.org/10.37236/1001">Higher Spin Alternating sign matrices</a>, El. J. Combinat 14 (2007) #R83
%F A143670 Apart from the trivial formulas |ASM(0, n)| = 1 (since ASM(0, n) contains only the n X n zero matrix), |ASM(1, r)| = 1 and |ASM(2, r)| = r+1, the only previously- known formula for a special case of |ASM(n, r)| is |ASM(n, 1)| = Sum_{i=0..n-1} (3*i+1)!/(n+1)!.
%e A143670 The array begins:
%e A143670 ========================================================
%e A143670 ....|.r=0|..r=1.|.....r=2.|.......r=3.|..........r=4.|
%e A143670 n=1.|..1.|...1..|......1..|.........1.|...........1..|.A000012
%e A143670 n=2.|..1.|...2..|......3..|.........4.|...........5..|.A000027
%e A143670 n=3.|..1.|...7..|.....26..|........70.|.........155..|
%e A143670 n=4.|..1.|..42..|....628..|......5102.|.......28005..|
%e A143670 n=5.|..1.|.429..|..41784..|...1507128.|....28226084..|
%e A143670 n=6.|..1.|7436..|7517457..|1749710096.|152363972022..|
%e A143670 ========================================================
%Y A143670 Cf. A000012, A000027, A005130.
%K A143670 more,nonn,tabl
%O A143670 1,5
%A A143670 _Jonathan Vos Post_, Aug 28 2008
%E A143670 Some terms of the 7th diagonal from _R. J. Mathar_, Mar 04 2010