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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).

%I A062734 #66 Apr 07 2023 16:20:26
%S A062734 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,
%T A062734 0,0,1296,3660,5700,6165,4945,2997,1365,455,105,15,1,0,0,0,0,0,0,
%U A062734 16807,68295,156555,258125,331506,343140,290745,202755,116175,54257,20349
%N 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).
%C A062734 T(n,n-1) = n^(n-2) counts free labeled trees A000272.
%C A062734 T(n,n) counts labeled connected unicyclic graphs A057500. - _Geoffrey Critzer_, Oct 07 2012
%D A062734 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
%D A062734 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.5.
%H A062734 Seiichi Manyama, <a href="/A062734/b062734.txt">Table of n, a(n) for n = 1..9919</a> (terms 1..75 from Alex Ermolaev, terms 76..175 from Alois P. Heinz)
%H A062734 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], 2017; Table 58.
%F A062734 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
%e A062734 Triangle starts:
%e A062734 [1],
%e A062734 [0, 1],
%e A062734 [0, 0, 3,  1],
%e A062734 [0, 0, 0, 16,  15,   6,   1],
%e A062734 [0, 0, 0,  0, 125, 222, 205, 120, 45, 10, 1],
%e A062734 ...
%t A062734 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 A062734 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 *)
%o A062734 (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 */
%Y A062734 Cf. A001187 (row sums), A054924 (unlabeled case), A061540 (a subdiagonal).
%Y A062734 See A123527 for another version (without leading zeros in each row).
%K A062734 easy,nonn,tabf
%O A062734 1,6
%A A062734 _Vladeta Jovovic_, Jul 12 2001