A053400 Number of 3-multigraphs on n nodes.

1, 4, 20, 276, 10688, 1601952, 892341888, 1799786093088, 13042490003160192, 341378170022783017472, 32526326484972756063585792, 11367103329997359707194173746176, 14669222110846093400698801891700529152

Offset: 1

Author

Vladeta Jovovic, Jan 06 2000

Keywords

• easy
• nonn
• nice

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY,1973.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Harald Fripertinger, The cycle type of the induced action on 2-subsets
• Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Crossrefs

Column k=3 of A063841.

Cf. A004102.

Programs

Python

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053400(n): return int(sum(Fraction(1<<(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())<<1),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024