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)).

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, 0, 0, 0, 2000, 11104, 33880, 73480, 123485, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1, 0, 0, 0, 0, 0, 41472, 337920, 1536000, 5062080

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; Table 76.

Formula

E.g.f.: 1+log( Sum_{n >= 0, k >= 0} binomial(n*(n-1), k)*x^n/n!*y^k ).

Example

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, 0, 0,   0, 2000, 11104, 33880,  73480, 123485, 166860, 184426, 167900, ...;
0, 0, 0,   0,    0, 41472, 337920,1536000,5062080,.. ;
0, 0, 0,   0,    0,     0, 1075648,...

Mathematica

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 *)

Prog

(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))}
{ for(n=0, 5, print(row(n))) } \\ Andrew Howroyd, Jan 11 2022

Crossrefs

Cf. A003027 (row sums), A054733 (unlabeled case), A057273 (strongly connected), A097629 (diagonal), A123554 (not necessarily connected).

Sequence in context: A057274 A088632 A057272 * A054547 A202717 A291195

Adjacent sequences: A062732 A062733 A062734 * A062736 A062737 A062738

Keyword

easy,nonn,tabf

Author

Vladeta Jovovic, Jul 12 2001

Status

approved