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A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.

%I A333361 #11 Mar 16 2020 16:27:53
%S A333361 1,1,0,1,0,0,1,1,0,0,1,1,2,0,0,1,1,5,2,0,0,1,1,6,10,3,0,0,1,1,6,20,24,
%T A333361 3,0,0,1,1,6,23,69,42,4,0,0,1,1,6,24,110,196,83,4,0,0,1,1,6,24,126,
%U A333361 427,554,132,5,0,0,1,1,6,24,129,603,1681,1368,222,5,0,0
%N A333361 Array read by antidiagonals: T(n,k) is the number of directed loopless multigraphs with n arcs and k vertices.
%H A333361 Andrew Howroyd, <a href="/A333361/b333361.txt">Table of n, a(n) for n = 0..1325</a> (antidiagonals 0..50)
%F A333361 T(n,k) = A052170(n) for k >= 2*n.
%e A333361 ==================================================
%e A333361 n\k | 0 1 2   3    4     5     6      7      8
%e A333361 ----+---------------------------------------------
%e A333361   0 | 1 1 1   1    1     1     1      1      1 ...
%e A333361   1 | 0 0 1   1    1     1     1      1      1 ...
%e A333361   2 | 0 0 2   5    6     6     6      6      6 ...
%e A333361   3 | 0 0 2  10   20    23    24     24     24 ...
%e A333361   4 | 0 0 3  24   69   110   126    129    130 ...
%e A333361   5 | 0 0 3  42  196   427   603    668    684 ...
%e A333361   6 | 0 0 4  83  554  1681  2983   3811   4116 ...
%e A333361   7 | 0 0 4 132 1368  5881 13681  20935  24979 ...
%e A333361   8 | 0 0 5 222 3240 19448 59680 112943 154504 ...
%e A333361   ...
%o A333361 (PARI) \\ here G(k,x) gives column k as rational function.
%o A333361 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 A333361 edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^(v[i]-1))}
%o A333361 G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i)); s/n!}
%o A333361 T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
%o A333361 {my(A=T(10)); for(n=1, #A, print(A[n, ]))}
%Y A333361 Columns k=0..4 are A000007, A000007, A008619, A037240, A050930.
%Y A333361 Cf. A052170, A138107, A192517, A290428.
%K A333361 nonn,tabl
%O A333361 0,13
%A A333361 _Andrew Howroyd_, Mar 16 2020