|
%I #29 Apr 30 2022 08:21:45
%S 1,0,1,0,0,1,1,0,0,0,2,2,1,1,0,0,0,0,3,5,5,4,2,1,1,0,0,0,0,0,6,13,19,
%T 22,20,14,9,5,2,1,1,0,0,0,0,0,0,11,33,67,107,132,138,126,95,64,40,21,
%U 10,5,2,1,1,0,0,0,0,0,0,0,23,89,236,486,814,1169,1454,1579,1515,1290,970,658,400,220,114
%N Triangle read by rows: T(n,k) = number of nonisomorphic unlabeled connected graphs with n nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%H R. W. Robinson, <a href="/A054924/b054924.txt">Rows 1 to 20 of triangle, flattened</a> (corrected by Sean A. Irvine, Apr 29 2022)
%H G. A. Baker et al., <a href="http://dx.doi.org/10.1103/PhysRev.164.800">High-temperature expansions for the spin-1/2 Heisenberg model</a>, Phys. Rev., 164 (1967), 800-817.
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a054/A054924.java">Java code</a> (github)
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/graphs/">Small graphs</a>
%H M. L. Stein and P. R. Stein, <a href="http://dx.doi.org/10.2172/4180737">Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points</a>. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967
%e Triangle begins:
%e 1;
%e 0,1;
%e 0,0,1,1;
%e 0,0,0,2,2,1,1;
%e 0,0,0,0,3,5,5,4,2,1,1;
%e 0,0,0,0,0,6,13,19,22,20,14,9,5,2,1,1;
%e the last batch giving the numbers of connected graphs with 6 nodes and from 0 to 15 edges.
%t A076263 gives a Mathematica program which produces the nonzero entries in each row.
%t Needs["Combinatorica`"]; Table[Print[row = Join[Array[0&, n-1], Table[ Count[ Combinatorica`ListGraphs[n, k], g_ /; Combinatorica`ConnectedQ[g]], {k, n-1, n*(n-1)/2}]]]; row, {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Jan 15 2015 *)
%Y Cf. A008406, A054925.
%Y Other versions of this triangle: A046751, A076263, A054923, A046742.
%Y Row sums give A001349, column sums give A002905. A046751 is essentially the same triangle. A054923 and A046742 give same triangle but read by columns.
%Y Main diagonal is A000055. Next diagonal is A001429. Largest entry in each row gives A001437.
%K nonn,easy,nice,tabf
%O 1,11
%A _N. J. A. Sloane_
|
