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 A318795 #14 Mar 29 2020 15:47:53
%S A318795 1,1,1,1,4,1,1,5,4,1,1,11,10,4,1,1,14,24,10,4,1,1,24,51,33,10,4,1,1,
%T A318795 30,114,78,33,10,4,1,1,45,219,224,91,33,10,4,1,1,55,424,549,277,91,33,
%U A318795 10,4,1,1,76,768,1403,792,298,91,33,10,4,1,1,91,1352,3292,2341,881,298,91,33,10,4,1
%N A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.
%H A318795 Andrew Howroyd, <a href="/A318795/b318795.txt">Table of n, a(n) for n = 1..1275</a>
%H A318795 Andrew Howroyd, <a href="/A318795/a318795.txt">Additional PARI Programs</a>
%F A318795 T(n,k) = T(k,k) for n > k.
%e A318795 Array begins:
%e A318795 ===========================================================
%e A318795 n\k| 1 2 3 4 5 6 7 8 9 10 11 12
%e A318795 ---+-------------------------------------------------------
%e A318795 1 | 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e A318795 2 | 1 4 5 11 14 24 30 45 55 76 91 119 ...
%e A318795 3 | 1 4 10 24 51 114 219 424 768 1352 2278 3759 ...
%e A318795 4 | 1 4 10 33 78 224 549 1403 3292 7677 16934 36581 ...
%e A318795 5 | 1 4 10 33 91 277 792 2341 6654 18802 51508 138147 ...
%e A318795 6 | 1 4 10 33 91 298 881 2825 8791 27947 87410 272991 ...
%e A318795 7 | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ...
%e A318795 8 | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ...
%e A318795 9 | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ...
%e A318795 ...
%t A318795 permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A318795 c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
%t A318795 M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
%t A318795 Table[M[n-k+1, n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Sep 12 2018, after _Andrew Howroyd_ *)
%o A318795 (PARI) \\ see also link.
%o A318795 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A318795 K(q, t, k)={1/prod(j=1, #q, (1-y^lcm(t,q[j]) + O(y*y^k))^gcd(t, q[j]))}
%o A318795 M(m, n, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(polcoef(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n), k)); s/m!}
%o A318795 for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated _Andrew Howroyd_, Mar 29 2020
%Y A318795 Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.
%Y A318795 Main diagonal is A007716.
%Y A318795 Cf. A214398, A246106, A304942, A318805.
%K A318795 nonn,tabl
%O A318795 1,5
%A A318795 _Andrew Howroyd_, Sep 03 2018