A290776 Triangle T(n,k) read by rows: the number of connected, loopless, non-oriented, vertex-labeled graphs with n >= 0 edges and k >= 1 vertices, allowing multi-edges.

1, 0, 1, 0, 1, 3, 0, 1, 7, 16, 0, 1, 12, 63, 125, 0, 1, 18, 162, 722, 1296, 0, 1, 25, 341, 2565, 10140, 16807, 0, 1, 33, 636, 7180, 47100, 169137, 262144, 0, 1, 42, 1092, 17335, 168285, 987567, 3271576, 4782969, 0, 1, 52, 1764, 37750, 509545, 4364017, 23315936, 72043092, 100000000

Offset: 0

Author

R. J. Mathar, Aug 10 2017

Keywords

• nonn
• tabl

Comments

This is the vertex-labeled companion to A191646.

Examples

			The triangle starts in row n=0 with 1 <= k <= n+1 vertices as
  1;
  0, 1;
  0, 1,  3;
  0, 1,  7,   16;
  0, 1, 12,   63,   125;
  0, 1, 18,  162,   722,   1296;
  0, 1, 25,  341,  2565,  10140,   16807;
  0, 1, 33,  636,  7180,  47100,  169137,   262144;
  0, 1, 42, 1092, 17355, 168285,  987567,  3271576,  4782969;
  0, 1, 52, 1764, 37750, 509545, 4364017, 23315936, 72043092, 100000000;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 62.

Crossrefs

Cf. A055998 (k=3), A000272 (diagonal), A060091 (k=4?), A060093 (k=5?).

Programs

Mathematica

S[m_, n_] := Binomial[Binomial[m, 2] + n - 1, n];
R[nn_] := Module[{cc = Array[0&, {nn, nn}]}, cc[[1, 1]] = 1; For[m = 1, m <= nn, m++, For[n = 1, n <= nn-1, n++, cc[[m, n+1]] = S[m, n] - S[m-1, n] - Sum[Sum[Binomial[m-1, i-1]*cc[[i, j+1]]*S[m-i, n-j], {j, 1, n}], {i, 2, m-1}]]]; cc // Transpose];
A = R[10];
Table[A[[n, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 13 2018, after Andrew Howroyd *)

PARI

\\ here S(m,n) is m nodes with n edges, not necessarily connected
S(m,n)={ binomial(binomial(m,2) + n - 1, n) }
R(N)={ my(C=matrix(N,N)); C[1,1]=1; for(m=1, N, for(n=1, N-1, C[m,n+1] = S(m,n) - S(m-1,n) - sum(i=2, m-1, sum(j=1, n, binomial(m-1, i-1)*C[i,j+1]*S(m-i, n-j))))); C~; }
{ my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n,k],", ")); print) } \\ Andrew Howroyd, May 13 2018

Extensions

Terms a(34) and beyond from Andrew Howroyd, May 13 2018