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 A333351 #17 Nov 30 2023 19:18:57
%S A333351 1,1,1,1,0,1,1,0,1,1,1,0,1,0,1,1,0,1,1,3,1,1,0,1,0,6,0,1,1,0,1,1,10,
%T A333351 22,15,1,1,0,1,0,15,0,130,0,1,1,0,1,1,21,158,760,822,105,1,1,0,1,0,28,
%U A333351 0,3355,0,6202,0,1,1,0,1,1,36,654,12043,93708,190050,52552,945,1,1,0,1,0,45,0,36935,0,3535448,0,499194,0,1
%N A333351 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n labeled nodes, n >= 0, k >= 0.
%H A333351 Andrew Howroyd, <a href="/A333351/b333351.txt">Table of n, a(n) for n = 0..405</a>
%e A333351 Array begins:
%e A333351 =================================================================
%e A333351 n\k | 0 1 2 3 4 5 6 7
%e A333351 ----+------------------------------------------------------------
%e A333351 0 | 1 1 1 1 1 1 1 1 ...
%e A333351 1 | 1 0 0 0 0 0 0 0 ...
%e A333351 2 | 1 1 1 1 1 1 1 1 ...
%e A333351 3 | 1 0 1 0 1 0 1 0 ...
%e A333351 4 | 1 3 6 10 15 21 28 36 ...
%e A333351 5 | 1 0 22 0 158 0 654 0 ...
%e A333351 6 | 1 15 130 760 3355 12043 36935 100135 ...
%e A333351 7 | 1 0 822 0 93708 0 3226107 0 ...
%e A333351 8 | 1 105 6202 190050 3535448 45163496 431400774 3270643750 ...
%e A333351 ...
%o A333351 (PARI)
%o A333351 MultigraphsByDegreeSeq(n, limit, ok)={
%o A333351 local(M=Map(Mat([0, 1])));
%o A333351 my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A333351 my(recurse(r, h, p, q, v, e) = if(!p, if(ok(x^e+q, r), acc(x^e+q, v)), my(i=poldegree(p), t=pollead(p)); self()(r, limit, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(k=1, min(t, (limit-e)\m), self()(r, if(k==t, limit, i+m-1), p-k*x^i, q+k*x^(i+m), binomial(t, k)*v, e+k*m)))));
%o A333351 for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, limit, src[i, 1], 0, src[i, 2], 0))); Mat(M);
%o A333351 }
%o A333351 T(n,k)={if((n%2&&k%2)||(n==1&&k>0), 0, vecsum(MultigraphsByDegreeSeq(n, k, (p,r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[,2]))}
%o A333351 { for(n=0, 8, for(k=0, 7, print1(T(n,k), ", ")); print) }
%Y A333351 Rows n=4..6 are A000217(n+1), A244868 (with interspersed zeros), A244878.
%Y A333351 Columns k=0..4 are A000012, A123023, A002137, A108243 (with interspersed zeros), A367497.
%Y A333351 Cf. A059441 (graphs), A333157, A333330 (unlabeled nodes), A333467 (with loops).
%K A333351 nonn,tabl
%O A333351 0,20
%A A333351 _Andrew Howroyd_, Mar 15 2020