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A350733 Triangle read by rows: T(n,k) is the number of oriented graphs on n unlabeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

%I A350733 #8 Jan 29 2022 22:31:00
%S A350733 1,1,1,1,1,1,3,2,1,1,4,10,12,10,4,1,1,4,13,41,78,131,144,107,50,12,1,
%T A350733 1,4,14,55,187,539,1292,2500,3817,4512,4112,2740,1274,376,56,1,1,4,14,
%U A350733 58,240,1009,3643,11815,32538,76145,149724,247329,340364,387834,361450,271177,159872,71320,22690,4604,456
%N A350733 Triangle read by rows: T(n,k) is the number of oriented graphs on n unlabeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.
%H A350733 Andrew Howroyd, <a href="/A350733/b350733.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)
%e A350733 Triangle begins:
%e A350733   [0] 1;
%e A350733   [1] 1;
%e A350733   [2] 1, 1;
%e A350733   [3] 1, 1, 3,  2;
%e A350733   [4] 1, 1, 4, 10, 12, 10,   4;
%e A350733   [5] 1, 1, 4, 13, 41, 78, 131, 144, 107, 50, 12;
%e A350733   ...
%o A350733 (PARI)
%o A350733 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 A350733 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 A350733 row(n)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); Vecrev(s/n!)}
%o A350733 { for(n=0, 6, print(row(n))) }
%Y A350733 Row sums are A001174.
%Y A350733 Cf. A350734.
%K A350733 nonn,tabf
%O A350733 0,7
%A A350733 _Andrew Howroyd_, Jan 13 2022