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%I A135813 #15 Jul 01 2023 08:27:56
%S A135813 1,0,127,279554,4585352445,358295150440964,100303980203191474555,
%T A135813 82605709118517742843295238,173237539725464803175622157326841,
%U A135813 828591383820135935294977528049328110600
%N A135813 Number of coincidence-free length n lists of 7-tuples with all numbers 1,...,n in tuple position k, for k=1..7.
%C A135813 a(n) enumerates (ordered) lists of n 7-tuples such that every number from 1 to n appears once at each of the seven tuple positions and the j-th list member is not the tuple (j,j,j,j,j,j,j), for every j=1,..,n. Called coincidence-free 7-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.
%D A135813 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=7.
%H A135813 G. C. Greubel, <a href="/A135813/b135813.txt">Table of n, a(n) for n = 0..90</a>
%F A135813 a(n) = Sum_{j=0,..,n}( ((-1)^(n-j))*binomial(n,j)*(j!)^7 ). See the Charalambides reference a(n)=B_{n,7}.
%e A135813 7-tuple combinatorics: a(1)=0 because the only list of 7-tuples composed of 1 is [(1,1,1,1,1,1,1)] and this is a coincidence for j=1.
%e A135813 7-tuple combinatorics: from the 2^7=128 possible 7-tuples 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 7-tuples obtained by interchanging 1 and 2 in the first member. E.g. one of the a(2)=2^7-1 =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.
%t A135813 Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^7, {k, 0, n}], {n,0,25}] (* _G. C. Greubel_, Nov 23 2016 *)
%Y A135813 Cf. A135812 (coincidence-free 6-tuples).
%K A135813 nonn,easy
%O A135813 0,3
%A A135813 _Wolfdieter Lang_, Jan 21 2008

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