A053467 Number of directed 2-multigraphs on n nodes.
1, 6, 138, 22815, 29197989, 286181094816, 21712697070199704, 12980080058620326927885, 62082385554465497895132149640, 2405193620328895144597707267893468286, 762399006478986275307113015668690102196187810
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..40
Programs
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Mathematica
Table[CycleIndex[PairGroup[SymmetricGroup[n], Ordered], t] /.Table[t[i] -> 1 + x^i + y^i, {i, 1, n^2}] /. {x -> 1, y -> 1}, {n, 1, 7}] (* Geoffrey Critzer, Mar 08 2011 *) permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m]; edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1]; a[n_] := (s=0; Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!); Array[a, 15] (* Jean-François Alcover, Jul 08 2018, after Andrew Howroyd *) -
PARI
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m} edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i]-1)} a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017 -
Python
from itertools import combinations from math import prod, gcd, factorial from fractions import Fraction from sympy.utilities.iterables import partitions def A053467(n): return int(sum(Fraction(3**((sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))<<1)+sum(q*r**2 for q, r in p.items())-s),prod(q**r*factorial(r) for q, r in p.items())) for s, p in partitions(n,size=True))) # Chai Wah Wu, Jul 10 2024
Extensions
a(11) from Andrew Howroyd, Oct 22 2017
Comments