A053418
Number of unlabeled directed graphs with n arcs and no isolated vertices.
Original entry on oeis.org
1, 1, 5, 17, 80, 365, 1981, 11222, 69511, 455663, 3169244, 23170347, 177513359, 1418920570, 11798710013, 101778754655, 908722427531, 8380602471646, 79692654473866, 780142956502644, 7851084073063731, 81120767066417308
Offset: 0
A053598
Number of n-node unlabeled digraphs without isolated nodes.
Original entry on oeis.org
1, 0, 2, 13, 202, 9390, 1531336, 880492496, 1792477159408, 13026163465206704, 341247403996148180800, 32522568124623933138617088, 11366712907916015518547782806784, 14669074325967499043636521641422216704, 70315641946149306808455637518883828774996992
Offset: 0
-
b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= n-> b(n$2, [])-`if`(n=0, 0, b(n-1$2, [])):
seq(a(n), n=0..16); # Alois P. Heinz, Sep 04 2019
-
Needs["Combinatorica`"];
nn=15;s=Sum[NumberOfDirectedGraphs[n]x^n,{n,0,nn}];CoefficientList[Series[s (1-x),{x,0,nn}],x] (* Geoffrey Critzer, Oct 09 2012 *)
Join[{1}, Table[GraphPolynomial[n, x, Directed] /. x -> 1, {n, 0, 15}] // Differences] (* Jean-François Alcover, Feb 04 2015 *)
-
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053598(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024
A054547
Triangular array giving number of labeled digraphs on n unisolated nodes and k=0..n*(n-1) arcs.
Original entry on oeis.org
0, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 12, 140, 435, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 240, 2520, 11604, 34150, 73560, 123495, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1
Offset: 1
Triangle T(n,k) begins:
[0],
[0,2,1],
[0,0,12,20,15,6,1],
[0,0,12,140,435,768,920,792,495,220,66,12,1],
...
-
row(n) = {Vecrev(sum(i=0, n, (-1)^(n-i)*binomial(n,i)*(1 + 'y)^(i*(i-1))), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022
Showing 1-3 of 3 results.
Comments