A053421 - OEIS

%I #16 Jul 10 2024 02:59:28

%S 1,6,56,2451,533358,661452084,4364646955812,152397092027960154,

%T 28427450083725134688228,28645398830642924774967347088,

%U 157458251108667629202718200130101672,4760428376101385226312810920945121043818096

%N Number of 5-multigraphs on n nodes.

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

%H Andrew Howroyd, <a href="/A053421/b053421.txt">Table of n, a(n) for n = 1..50</a>

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_42.html">The cycle type of the induced action on 2-subsets</a>

%H Vladeta Jovovic, <a href="/A063843/a063843.rtf">Formulae for the number T(n,k) of n-multigraphs on k nodes</a>

%o (Python)

%o from itertools import combinations

%o from math import prod, gcd, factorial

%o from fractions import Fraction

%o from sympy.utilities.iterables import partitions

%o 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

%Y Column k=5 of A063841.

%Y Cf. A000088, A004102, A053400.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Jan 11 2000

%E a(12) from _Andrew Howroyd_, Oct 22 2017