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%I #35 Sep 08 2022 08:45:30
%S 1,27,343,3375,29791,250047,2048383,16581375,133432831,1070599167,
%T 8577357823,68669157375,549554511871,4397241253887,35181150961663,
%U 281462092005375,2251748274470911,18014192351838207,144114363443707903
%N Number of n-tuples where each entry is chosen from the subsets of {1,2,3} such that the intersection of all n entries is empty.
%C The general formula where each entry is chosen from the subsets of {1,..,k} is (2^n-1)^k. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n-1)^k, namely the set of all k-tuples with each entry chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for of the k entries {1,..,n} is forbidden. The bijection is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i. Sequence A060867 is the case where the entries are chosen from subsets of {1,2}.
%D R. P. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth & Brooks 1986 p. 11.
%H Vincenzo Librandi, <a href="/A128831/b128831.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).
%F a(n) = (2^n-1)^3.
%F G.f.: x*(8*x^2+12*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)). [_Colin Barker_, Nov 17 2012]
%e a(1)=(2^1-1)^3=1 because only one tuple of length one, namely ({}) has an empty intersection of its sole entry.
%e a(2)=27 because the valid 2-tuples are: ({},{}), ({},{1}), ({},{2}), ({},{3}), ({},{1,2}), ({},{1,3}), ({},{2,3}), ({},{1,2,3}), ({1},{}), ({2},{}), ({3},{}), ({1,2},{}), ({1,3},{}), ({2,3},{}), ({1,2,3},{}), ({1},{2}), ({1},{3}), ({1},{2,3}), ({2},{1}), ({2},{3}), ({2},{1,3}), ({3},{1}), ({3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,3},{2}), ({2,3},{1})
%p for k from 1 to 20 do (2^k-1)^3; od;
%t Table[(2^n - 1)^3, {n, 30}] (* _Vincenzo Librandi_, Mar 04 2018 *)
%o (Magma) [(2^n-1)^3: n in [1..20]]; // _Vincenzo Librandi_, Mar 04 2018
%o (PARI) a(n) = (2^n-1)^3; \\ _Altug Alkan_, Mar 04 2018
%Y Cf. A060867.
%K easy,nonn
%O 1,2
%A Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
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