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A309936 Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

%I A309936 #35 Jan 11 2021 17:00:30
%S A309936 0,1,0,1,1,1,0,1,2,3,1,1,0,1,3,7,6,4,1,1,0,1,4,13,17,17,8,4,1,1,0,1,6,
%T A309936 25,44,56,41,24,9,4,1,1,0,1,7,40,101,164,158,117,57,26,9,4,1,1,0,1,9,
%U A309936 65,216,450,562,503,315,162,64,27,9,4,1,1
%N A309936 Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.
%C A309936 Covering k vertices means there are no vertices of degree zero.
%H A309936 Andrew Howroyd, <a href="/A309936/b309936.txt">Table of n, a(n) for n = 1..650</a> (rows 1..25)
%F A309936 T(n,k) = A192517(k,n) - A192517(k-1,n) for k > 1.
%e A309936 Triangle begins:
%e A309936   0, 1;
%e A309936   0, 1, 1,  1;
%e A309936   0, 1, 2,  3,   1,   1;
%e A309936   0, 1, 3,  7,   6,   4,   1,   1;
%e A309936   0, 1, 4, 13,  17,  17,   8,   4,  1,  1;
%e A309936   0, 1, 6, 25,  44,  56,  41,  24,  9,  4, 1, 1;
%e A309936   0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1;
%e A309936   ...
%o A309936 (PARI)
%o A309936 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 A309936 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)*if(c%2, 1, t(c/2)))}
%o A309936 C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
%o A309936 T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
%o A309936 { my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }
%Y A309936 Row sums are A050535.
%Y A309936 Columns k=3..4 are A253186, A328652.
%Y A309936 Cf. A191646, A192517, A327615.
%K A309936 nonn,tabf
%O A309936 1,9
%A A309936 _Andrew Howroyd_, Oct 23 2019