

A135811


Number of coincidencefree length n lists of 5tuples with all numbers 1,...,n in tuple position k, for k=1..5.


3



1, 0, 31, 7682, 7931709, 24843464324, 193342583284315, 3250662144028779654, 106536051676371091349113, 6291424280473807580386161416, 629175403160580417773688864819351
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OFFSET

0,3


COMMENTS

a(n) enumerates (ordered) lists of n 5tuples such that every number from 1 to n appears once at each of the five tuple positions and the jth list member is not the tuple (j,j,j,j,j), for every j=1,..,n. Called coincidencefree 5tuple 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)^(nj))*binomial(n,j)*(j!)^5 ). See the Charalambides reference a(n)=B_{n,5}.


EXAMPLE

5tuple combinatorics: a(1)=0 because the only list of 5tuples composed of 1 is [(1,1,1,1,1)] and this is a coincidence for j=1.
5tuple combinatorics: from the 2^5 possible 5tuples 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 5tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^51 =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

Cf. A135810 (coincidencefree 4tuples). A135812 (coincidencefree 6tuples).
Sequence in context: A212858 A059384 A136676 * A119598 A139295 A261947
Adjacent sequences: A135808 A135809 A135810 * A135812 A135813 A135814


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Jan 21 2008


STATUS

approved



