A053420 Number of 4-multigraphs on n nodes.

1, 5, 35, 900, 90005, 43571400, 95277592625, 925609100039625, 40119721052610123750, 7833164300852979350336250, 6953552738579427778531249187500, 28293472829338822230349054996265275000, 531350037528849507720092485196308155336875000

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=4 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 A053420(n): return int(sum(Fraction(5**(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

Terms a(12) and beyond from Andrew Howroyd, Oct 22 2017