A191646 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.
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, 37, 85, 110, 70, 23, 0, 1, 9, 61, 193, 348, 339, 185, 47, 0, 1, 11, 95, 396, 969, 1318, 1067, 479, 106, 0, 1, 13, 141, 771, 2445, 4457, 4940, 3294, 1279, 235
Offset: 0
Examples
Triangle T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows: 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; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..119 from R. J. Mathar)
- R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; see Section 4.
- Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
- B. D. McKay and A. Piperno, Practical Graph Isomorphism, II, J. Symbolic Computation 60 (2013), 94-112.
- Gordon Royle, Small Multigraphs.
- Gus Wiseman, Illustration of the 33 connected multigraphs counted in row 5.
Crossrefs
Programs
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PARI
EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)} 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))} 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} 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)))} G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!} R(n)={Mat(apply(p->Col(p+O(y^n),-n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))} { my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k], ", "));print) } \\ Andrew Howroyd, May 14 2018