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 A158345 #9 Jun 29 2025 21:56:54
%S A158345 1,5,13,53,61,845,253,7509,16141,128045,4093,1785965,16381,23576285,
%T A158345 55921333,274696789,262141,5338300157,1048573,63028146573,
%U A158345 117924207421,995274180125,16777213,15265519672173,14283159085861
%N A158345 The number of pairs of independent outcomes when rolling an n-sided die. Or in other words, the number of pairs of proper subsets A,B of a set S, such that #A/#S * #B/#S = #(A \intersect B)/#S.
%H A158345 <a href="http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_cs;action=display;num=1234635667">wwu riddle forum thread on the problem</a>
%e A158345 For N=4 we have 53 solutions, because {1,2,3,4} together with any proper subset yields 2*15-1 = 29 valid pairs, and a further 24 pairs can be obtained from {1,2} & {1,3}, by substituting the numbers with any permutation of (1,2,3,4).
%t A158345 Sum[Total[s!/(c!(#-c)!(s c/#-c)!(s - # - s c/# + c)!) &/@Select[Divisors[s c], c <= # <= s &]], {c,1,s}] (* "Eigenray", see link, Feb 15 2009 *)
%K A158345 nonn
%O A158345 1,2
%A A158345 Harmen Wassenaar (towr(AT)ai.rug.nl), Mar 16 2009