A062734 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).

1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 16, 15, 6, 1, 0, 0, 0, 0, 125, 222, 205, 120, 45, 10, 1, 0, 0, 0, 0, 0, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 0, 0, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349

Offset: 1

Author

Vladeta Jovovic, Jul 12 2001

Keywords

• easy
• nonn
• tabf

Comments

T(n,n-1) = n^(n-2) counts free labeled trees A000272.

T(n,n) counts labeled connected unicyclic graphs A057500. - Geoffrey Critzer, Oct 07 2012

Examples

			Triangle starts:
[1],
[0, 1],
[0, 0, 3,  1],
[0, 0, 0, 16,  15,   6,   1],
[0, 0, 0,  0, 125, 222, 205, 120, 45, 10, 1],
...
		

References

• Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - N. J. A. Sloane, Apr 06 2012
• F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.5.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..9919 (terms 1..75 from Alex Ermolaev, terms 76..175 from Alois P. Heinz)
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; Table 58.

Crossrefs

Cf. A001187 (row sums), A054924 (unlabeled case), A061540 (a subdiagonal).

See A123527 for another version (without leading zeros in each row).

Programs

Mathematica

nn=6;s=Sum[(1+y)^Binomial[n,2] x^n/n!,{n,0,nn}]; Range[0,nn]!CoefficientList[Series[Log[ s]+1,{x,0,nn}],{x,y}]//Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, k]]; (* Michael Somos, Aug 12 2017 *)

PARI

{T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( log( sum(m=0, n, (1 + y)^(m * (m-1)/2) * x^m/m!)), n), k))}; /* Michael Somos, Aug 12 2017 */

Formula

G.f.: Sum_{n>=1, k>=0} T(n, k) * x^n/n! * y^k = log(Sum_{n>=0} (1 + y)^binomial(n, 2) * x^n/n!). - Ralf Stephan, Jan 18 2005