

A135813


Number of coincidencefree length n lists of 7tuples with all numbers 1,...,n in tuple position k, for k=1..7.


3



1, 0, 127, 279554, 4585352445, 358295150440964, 100303980203191474555, 82605709118517742843295238, 173237539725464803175622157326841, 828591383820135935294977528049328110600
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OFFSET

0,3


COMMENTS

a(n) enumerates (ordered) lists of n 7tuples such that every number from 1 to n appears once at each of the seven tuple positions and the jth list member is not the tuple (j,j,j,j,j,j,j), for every j=1,..,n. Called coincidencefree 7tuple 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=7.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..90


FORMULA

a(n) = Sum_{j=0,..,n}( ((1)^(nj))*binomial(n,j)*(j!)^7 ). See the Charalambides reference a(n)=B_{n,7}.


EXAMPLE

7tuple combinatorics: a(1)=0 because the only list of 7tuples composed of 1 is [(1,1,1,1,1,1,1)] and this is a coincidence for j=1.
7tuple combinatorics: from the 2^7=128 possible 7tuples of numbers 1 and 2 all except (1,1,1,1,1,1,1) appear as first members of the length 2 lists. The second members are the 7tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^71 =127 lists is [(1,1,1,1,1,1,2),(2,2,2,2,2,2,1)]. The list [(1,1,1,1,1,1,1),(2,2,2,2,2,2,2) 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!)^7, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Nov 23 2016 *)


CROSSREFS

Cf. A135812 (coincidencefree 6tuples).
Sequence in context: A195218 A215692 A212860 * A112016 A263165 A135982
Adjacent sequences: A135810 A135811 A135812 * A135814 A135815 A135816


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang Jan 21 2008


STATUS

approved



