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%I #15 Jul 10 2024 02:59:16
%S 1,4,20,276,10688,1601952,892341888,1799786093088,13042490003160192,
%T 341378170022783017472,32526326484972756063585792,
%U 11367103329997359707194173746176,14669222110846093400698801891700529152
%N Number of 3-multigraphs on n nodes.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY,1973.
%H Andrew Howroyd, <a href="/A053400/b053400.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 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
%Y Column k=3 of A063841.
%Y Cf. A004102.
%K easy,nonn,nice
%O 1,2
%A _Vladeta Jovovic_, Jan 06 2000