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%I #10 Jul 03 2023 08:32:18
%S 3,27,147,675,2883,11907,48387,195075,783363,3139587,12570627,
%T 50307075,201277443,805208067,3221028867,12884508675,51538821123,
%U 206156857347,824630575107,3298528591875,13194126950403,52776532967427
%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 contains exactly one element.
%C There is the following general formula: The number T(n,k,r) of n-tuples where each entry is chosen from the subsets of {1,2,..,k} such that the intersection of all n entries contains exactly r elements is: T(n,k,r) = C(k,r) * (2^n - 1)^(k-r). This may be shown by exhibiting a bijection to a set whose cardinality is obviously C(k,r) * (2^n - 1)^(k-r), namely the set of all k-tuples where each entry is chosen from subsets of {1,..,n} in the following way: Exactly r entries must be {1,..,n} itself (there are C(k,r) ways to choose them) and the remaining (k-r) entries must be chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for each of the (k-r) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n - 1)^(k-r) possibilities to do that, hence the formula). The bijection into this set 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.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, -14, 8).
%F a(n) = 3*(2^n-1)^2.
%F G.f.: 3*x*(1+2*x)/(1-7*x+14*x^2-8*x^3). [_Colin Barker_, Feb 08 2012]
%e a(1)=3 because the three sequences of length one are: ({1}), ({2}), ({3}).
%e a(2)=27 because the twenty-seven sequences of length two are:
%e ({1},{1}), ({2},{2}), ({3},{3}), ({1},{1,2}),
%e ({1},{1,3}), ({2},{1,2}), ({2},{2,3}), ({3},{1,3}),
%e ({3},{2,3}), ({1,2},{1}), ({1,3},{1}), ({1,2},{2}),
%e ({2,3},{2}), ({1,3},{3}), ({2,3},{3}), ({1},{1,2,3}),
%e ({2},{1,2,3}), ({3},{1,2,3}), ({1,2,3},{1}), ({1,2,3},{2}),
%e ({1,2,3},{3}), ({1,2},{1,3}), ({1,3},{1,2}), ({1,2},{2,3}),
%e ({2,3},{1,2}), ({1,3},{2,3}), ({2,3},{1,3}).
%p for k from 1 to 41 do 3*(2^k-1)^2; od;
%Y Cf. A128831, A128832, A128833, A095121.
%K nonn
%O 1,1
%A Peter C. Heinig (algorithms(AT)gmx.de), Apr 17 2007
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