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A135809 Number of coincidence-free length n lists of 3-tuples with all numbers 1,...,n in tuple position k, for k=1,2,3.

%I A135809 #19 Jul 01 2023 08:27:46
%S A135809 1,0,7,194,13005,1660964,363083155,125447139558,64534483387801,
%T A135809 47199368682436040,47309812970969661471,63078455495155600593290,
%U A135809 109143265990975402533003877,240033842542243124391262433004
%N A135809 Number of coincidence-free length n lists of 3-tuples with all numbers 1,...,n in tuple position k, for k=1,2,3.
%C A135809 a(n) enumerates (ordered) lists of n three-tuples such that every number from 1 to n appears once at each of the three tuple positions and the j-th list member is not the tuple (j,j,j), for every j=1,..,n. Called coincidence-free 3-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
%D A135809 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=3.
%H A135809 G. C. Greubel, <a href="/A135809/b135809.txt">Table of n, a(n) for n = 0..181</a>
%F A135809 a(n) = Sum_{j=0,..,n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^3 ). See the Charalambides reference a(n) = B_{n,3}.
%F A135809 a(n) ~ n!^3 ~ (2*Pi)^(3/2) * n^(3*n + 3/2) / exp(3*n). - _Vaclav Kotesovec_, Nov 19 2016
%e A135809 3-tuple combinatorics: a(1)=0 because the only list of 3-tuples with numbers 1 is [(1,1,1)] and this is a coincidence for j=1.
%e A135809 3-tuple combinatorics: the a(2)=7 coincidence-free 3-tuple lists of length n=2 are [(1,1,2),(2,2,1)], [(1,2,1),(2,1,2)], [(2,1,1),(1,2,2)], [(2,2,1), (1,1,2)], [(2,1,2),(1,2,1)], [(1,2,2),(2,1,1)] and [(2,2,2),(1,1, 1)]. The list [(1,1,1),(2,2,2)] has in fact two coincidences (j=1 and j=2).
%t A135809 Table[Sum[(-1)^k Binomial[n, k](n-k)!^3, {k, 0, n}], {n, 0, 13}] (* _Geoffrey Critzer_, Jun 17 2013 *)
%o A135809 (PARI) a(n)=sum(k=0,n,(-1)^k*binomial(n, k)*(n-k)!^3) \\ _Charles R Greathouse IV_, Nov 17 2016
%Y A135809 Cf. A089041 (coincidence-free 2-tuples), A135810 (coincidence-free 4-tuples).
%K A135809 nonn,easy
%O A135809 0,3
%A A135809 _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008, May 21 2008