OFFSET
0,3
COMMENTS
a(n) enumerates (ordered) lists of n 5-tuples such that every number from 1 to n appears once at each of the five tuple positions and the j-th list member is not the tuple (j,j,j,j,j), for every j=1,..,n. Called coincidence-free 5-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
REFERENCES
Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=5.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Sum_{j=0..n} ( ((-1)^(n-j))*binomial(n,j)*(j!)^5 ). See the Charalambides reference a(n)=B_{n,5}.
EXAMPLE
5-tuple combinatorics: a(1)=0 because the only list of 5-tuples composed of 1 is [(1,1,1,1,1)] and this is a coincidence for j=1.
5-tuple combinatorics: from the 2^5 possible 5-tuples of numbers 1 and 2 all except (1,1,1,1,1) appear as first members of the length 2 lists. The second members are the 5-tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^5-1 =31 lists is [(1,1,1,1,2),(2,2,2,2,1)]. The list [(1,1,1,1,1),(2,2,2,2,2) does not qualify because it has in fact two coincidences, those for j=1 and j=2.
MATHEMATICA
Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^5, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Nov 23 2016 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 21 2008
STATUS
approved