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.

1, 0, 7, 194, 13005, 1660964, 363083155, 125447139558, 64534483387801, 47199368682436040, 47309812970969661471, 63078455495155600593290, 109143265990975402533003877, 240033842542243124391262433004

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008, May 21 2008

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
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).
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=3.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..181

Crossrefs

Cf. A089041 (coincidence-free 2-tuples), A135810 (coincidence-free 4-tuples).

Programs

Mathematica

Table[Sum[(-1)^k Binomial[n, k](n-k)!^3, {k, 0, n}], {n, 0, 13}] (* Geoffrey Critzer, Jun 17 2013 *)

PARI

a(n)=sum(k=0,n,(-1)^k*binomial(n, k)*(n-k)!^3) \\ Charles R Greathouse IV, Nov 17 2016

Formula

a(n) = Sum_{j=0,..,n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^3 ). See the Charalambides reference a(n) = B_{n,3}.

a(n) ~ n!^3 ~ (2*Pi)^(3/2) * n^(3*n + 3/2) / exp(3*n). - Vaclav Kotesovec, Nov 19 2016