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

0,9

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.

Formula

T(n,k=3) = A253186(n) = A034253(n,k=2) for n >= 1. - Petros Hadjicostas, Oct 02 2019

Example

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;
  ...

Prog

(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

Crossrefs

Row sums give A076864. Diagonal is A000055.

Cf. A034253, A054923, A192517, A253186 (column k=3), A290778 (column k=4).

Cf. A000664, A007718, A036250, A050535, A191646, A191970, A275421, A317672, A322114, A322133, A322152.

Sequence in context: A384976 A144064 A172236 * A385017 A297321 A277938

Adjacent sequences: A191643 A191644 A191645 * A191647 A191648 A191649

Keyword

nonn,tabl,changed

Author

Alberto Tacchella, Jul 04 2011

Status

approved