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A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

%I A350734 #9 Jan 29 2022 15:00:04
%S A350734 1,0,1,0,0,3,2,0,0,0,8,12,10,4,0,0,0,0,27,68,127,144,107,50,12,0,0,0,
%T A350734 0,0,91,395,1144,2393,3767,4500,4112,2740,1274,376,56,0,0,0,0,0,0,350,
%U A350734 2170,9139,28606,71583,145600,244589,339090,387458,361394,271177,159872,71320,22690,4604,456
%N A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.
%H A350734 Andrew Howroyd, <a href="/A350734/b350734.txt">Table of n, a(n) for n = 1..1350</a> (rows 1..20)
%e A350734 Triangle begins:
%e A350734   [1] 1;
%e A350734   [2] 0, 1;
%e A350734   [3] 0, 0, 3, 2;
%e A350734   [4] 0, 0, 0, 8, 12, 10,   4;
%e A350734   [5] 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12;
%e A350734   ...
%o A350734 (PARI)
%o A350734 InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}
%o A350734 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 A350734 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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2))}
%o A350734 G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); s/n!}
%o A350734 row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n))}
%o A350734 { for(n=1, 6, print(row(n))) }
%Y A350734 Row sums are A086345.
%Y A350734 Column sums are A350915.
%Y A350734 Leading diagonal is A000238.
%Y A350734 The labeled version is A350732.
%Y A350734 Cf. A054733, A350733, A350750, A350914 (transpose).
%K A350734 nonn,tabf
%O A350734 1,6
%A A350734 _Andrew Howroyd_, Jan 13 2022