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%I #70 Mar 23 2023 23:08:39
%S 1,0,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,11,11,6,0,1,6,22,34,29,11,0,1,7,
%T 37,85,110,70,23,0,1,9,61,193,348,339,185,47,0,1,11,95,396,969,1318,
%U 1067,479,106,0,1,13,141,771,2445,4457,4940,3294,1279,235
%N Triangle read by rows: T(n,k) = number of connected multigraphs with n >= 0 edges and 1 <= k <= n+1 vertices, with no loops allowed.
%H Andrew Howroyd, <a href="/A191646/b191646.txt">Table of n, a(n) for n = 0..1274</a> (terms 0..119 from R. J. Mathar)
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], 2017; see Section 4.
%H Brendan McKay and Adolfo Piperno, <a href="http://users.cecs.anu.edu.au/~bdm/nauty/">nauty and Traces</a>. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
%H B. D. McKay and A. Piperno, <a href="http://dx.doi.org/10.1016/j.jsc.2013.09.003">Practical Graph Isomorphism, II</a>, J. Symbolic Computation 60 (2013), 94-112.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/graphs/multigraphs/">Small Multigraphs</a>.
%H Gus Wiseman, <a href="/A191646/a191646.png">Illustration of the 33 connected multigraphs counted in row 5</a>.
%F T(n,k=3) = A253186(n) = A034253(n,k=2) for n >= 1. - _Petros Hadjicostas_, Oct 02 2019
%e Triangle T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 2;
%e 0, 1, 3, 5, 3;
%e 0, 1, 4, 11, 11, 6;
%e 0, 1, 6, 22, 34, 29, 11;
%e ...
%o (PARI)
%o EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
%o InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
%o 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 edges(v,x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t-1)\2)*x^t + if(t%2,0,x^(t/2)))}
%o G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
%o R(n)={Mat(apply(p->Col(p+O(y^n),-n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
%o { my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k], ", "));print) } \\ _Andrew Howroyd_, May 14 2018
%Y Row sums give A076864. Diagonal is A000055.
%Y Cf. A034253, A054923, A192517, A253186 (column k=3), A290778 (column k=4).
%Y Cf. A000664, A007718, A036250, A050535, A191646, A191970, A275421, A317672, A322114, A322133, A322152.
%K nonn,tabl
%O 0,9
%A _Alberto Tacchella_, Jul 04 2011
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