This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135811 #17 Jan 30 2025 05:21:01
%S A135811 1,0,31,7682,7931709,24843464324,193342583284315,3250662144028779654,
%T A135811 106536051676371091349113,6291424280473807580386161416,
%U A135811 629175403160580417773688864819351,101332539752812925263300043667901615370,25215284734588360186537964188340607229390261
%N A135811 Number of coincidence-free length n lists of 5-tuples with all numbers 1..n in tuple position k, for k=1..5.
%C A135811 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.
%D A135811 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=5.
%H A135811 G. C. Greubel, <a href="/A135811/b135811.txt">Table of n, a(n) for n = 0..100</a>
%F A135811 a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^5. See the Charalambides reference a(n)=B_{n,5}.
%e A135811 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.
%e A135811 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.
%t A135811 Table[Sum[(-1)^(n - k)*Binomial[n, k]*(k!)^5, {k, 0, n}], {n,0,25}] (* _G. C. Greubel_, Nov 23 2016 *)
%Y A135811 Cf. A135810 (coincidence-free 4-tuples). A135812 (coincidence-free 6-tuples).
%K A135811 nonn,easy
%O A135811 0,3
%A A135811 _Wolfdieter Lang_, Jan 21 2008