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 A195644 #12 May 17 2020 03:55:27
%S A195644 1,1,1,1,1,1,1,1,3,1,1,1,3,15,1,1,1,3,19,199,1,1,1,3,19,379,6247,1,1,
%T A195644 1,3,19,391,22506,505623,1,1,1,3,19,391,25428,4063437,105997283,1,1,1,
%U A195644 3,19,391,25532,5422820,2303397986,58923059879,1,1,1,3,19,391,25532,5536654
%N A195644 T(n,k) is the number of lower triangles of an n X n 0..k array with all row sums equal to the length of the row and all column sums equal to the length of the column.
%H A195644 R. H. Hardin, <a href="/A195644/b195644.txt">Table of n, a(n) for n = 1..102</a>
%F A195644 T(n,k) = T(n,k-1) for k >= n, n >= 2. - _Andrew Howroyd_, May 16 2020
%e A195644 Table starts:
%e A195644 1 1 1 1 1 1 ...
%e A195644 1 1 1 1 1 1 ...
%e A195644 1 3 3 3 3 3 ...
%e A195644 1 15 19 19 19 19 ...
%e A195644 1 199 379 391 391 391 ...
%e A195644 1 6247 22506 25428 25532 25532 ...
%e A195644 1 505623 4063437 5422820 5536654 5539434 ...
%e A195644 1 105997283 2303397986 3868544673 4102276124 4116036800 ...
%e A195644 ...
%e A195644 Some solutions for n=5, k=4:
%e A195644 ..1..........1..........1..........1..........1..........1..........1
%e A195644 ..1.1........1.1........2.0........2.0........2.0........2.0........2.0
%e A195644 ..3.0.0......3.0.0......0.1.2......2.1.0......0.1.2......2.0.1......1.2.0
%e A195644 ..0.1.1.2....0.1.1.2....1.1.0.2....0.1.1.2....1.1.0.2....0.0.2.2....1.2.1.0
%e A195644 ..0.2.2.0.1..0.2.2.0.1..1.2.1.0.1..0.2.2.0.1..1.2.1.0.1..0.4.0.0.1..0.0.2.2.1
%o A195644 (PARI) \\ adapted from program for A257493.
%o A195644 T(n, k)={
%o A195644 local(M=Map(Mat([0, 1])));
%o A195644 my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A195644 my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, min(k, h-i), for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
%o A195644 for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1] + x^(r-1), 0, src[i, 2], r))); vecsum(Mat(M)[, 2])
%o A195644 } \\ _Andrew Howroyd_, May 16 2020
%Y A195644 Columns 1..7 are A000012, A195515, A195639, A195640, A195641, A195642, A195643.
%Y A195644 Main diagonal is A195638.
%Y A195644 Cf. A195522, A257493.
%K A195644 nonn,tabl
%O A195644 1,9
%A A195644 _R. H. Hardin_, Sep 21 2011