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A123527 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

%I A123527 #59 Mar 14 2024 18:12:13
%S A123527 1,1,3,1,16,15,6,1,125,222,205,120,45,10,1,1296,3660,5700,6165,4945,
%T A123527 2997,1365,455,105,15,1,16807,68295,156555,258125,331506,343140,
%U A123527 290745,202755,116175,54257,20349,5985,1330,210,21,1,262144,1436568
%N A123527 Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, n-1 <= k <= n(n-1)/2).
%D A123527 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). - From _N. J. A. Sloane_, Apr 06 2012
%D A123527 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
%H A123527 Seiichi Manyama, <a href="/A123527/b123527.txt"> Rows n = 1..40, flattened</a> (rows 1..14 from R. W. Robinson, rows 15..20 from Henrique G. G. Pereira)
%H A123527 T. Yanagita, T. Ichinomiya, <a href="https://arxiv.org/abs/1409.1979">Thermodynamic Characterization of Synchronization-Optimized Oscillator-Networks</a>, arXiv preprint arXiv:1409.1979 [nlin.AO], 2014.
%F A123527 For k >= C(n-1, 2) + 1 (not smaller!), T(n,k) = C(C(n,2),k) where C(n,k) is the binomial coefficient. See A084546. - _Geoffrey Critzer_, Dec 08 2011
%e A123527 Triangle begins:
%e A123527 n = 1
%e A123527   k = 0: 1
%e A123527   ****** total(1) = 1
%e A123527 n = 2
%e A123527   k = 1: 1
%e A123527   ****** total(2) = 1
%e A123527 n = 3
%e A123527   k = 2: 3
%e A123527   k = 3: 1
%e A123527   ****** total(3) = 4
%e A123527 n = 4
%e A123527   k = 3: 16
%e A123527   k = 4: 15
%e A123527   k = 5:  6
%e A123527   k = 6:  1
%e A123527   ****** total(4) = 38
%e A123527 n = 5
%e A123527   k = 4: 125
%e A123527   k = 5: 222
%e A123527   k = 6: 205
%e A123527   k = 7: 120
%e A123527   k = 8:  45
%e A123527   k = 9:  10
%e A123527   k = 10:  1
%e A123527   ****** total(5) = 728
%t A123527 nn = 8; a = Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 0, nn}]; f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Log[a], {x, 0, nn}], {x, y}],1]]] (* _Geoffrey Critzer_, Dec 08 2011 *)
%t A123527 T[ n_, k_] := If[ n < 1, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, n - 1 + k]]; (* _Michael Somos_, Aug 12 2017 *)
%Y A123527 See A062734 for another version with more information. Row sums give A001187.
%K A123527 nonn,tabf
%O A123527 1,3
%A A123527 _N. J. A. Sloane_, Nov 13 2006