A053421 Number of 5-multigraphs on n nodes.

1, 6, 56, 2451, 533358, 661452084, 4364646955812, 152397092027960154, 28427450083725134688228, 28645398830642924774967347088, 157458251108667629202718200130101672, 4760428376101385226312810920945121043818096

Offset: 1

Author

Vladeta Jovovic, Jan 11 2000

Keywords

• easy
• nonn

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=5 of A063841.

Cf. A000088, A004102, A053400.

Programs

Python

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053421(n): return int(sum(Fraction(6**(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())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

Extensions

a(12) from Andrew Howroyd, Oct 22 2017