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A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).

%I A062735 #28 Jan 11 2022 19:54:24
%S A062735 1,0,2,1,0,0,12,20,15,6,1,0,0,0,128,432,768,920,792,495,220,66,12,1,0,
%T A062735 0,0,0,2000,11104,33880,73480,123485,166860,184426,167900,125965,
%U A062735 77520,38760,15504,4845,1140,190,20,1,0,0,0,0,0,41472,337920,1536000,5062080
%N A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).
%H A062735 Andrew Howroyd, <a href="/A062735/b062735.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)
%H A062735 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], 2017; Table 76.
%F A062735 E.g.f.: 1+log( Sum_{n >= 0, k >= 0} binomial(n*(n-1), k)*x^n/n!*y^k ).
%e A062735 1;
%e A062735 0, 2, 1;
%e A062735 0, 0, 12, 20,   15,    6,      1;
%e A062735 0, 0, 0, 128,  432,  768,    920,    792,    495,    220,     66,    12, 1;
%e A062735 0, 0, 0,   0, 2000, 11104, 33880,  73480, 123485, 166860, 184426, 167900, ...;
%e A062735 0, 0, 0,   0,    0, 41472, 337920,1536000,5062080,.. ;
%e A062735 0, 0, 0,   0,    0,     0, 1075648,...
%t A062735 nn=7;s=Sum[(1+y)^(n^2-n) x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[ s]+1,{x,0,nn}],{x,y}]//Grid  (* returns triangle indexed from n = 0, _Geoffrey Critzer_, Oct 07 2012 *)
%o A062735 (PARI) row(n)={Vecrev(n!*polcoef(1 + log(sum(k=0, n, (1+y)^(k*(k-1))*x^k/k!, O(x*x^n))), n))}
%o A062735 { for(n=0, 5, print(row(n))) } \\ _Andrew Howroyd_, Jan 11 2022
%Y A062735 Cf. A003027 (row sums), A054733 (unlabeled case), A057273 (strongly connected), A097629 (diagonal), A123554 (not necessarily connected).
%K A062735 easy,nonn,tabf
%O A062735 1,3
%A A062735 _Vladeta Jovovic_, Jul 12 2001