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A128833 Number of n-tuples where each entry is chosen from the subsets of {1,2,3,4,5} such that the intersection of all n entries is empty.

%I A128833 #18 Mar 19 2024 08:31:02
%S A128833 1,243,16807,759375,28629151,992436543,33038369407,1078203909375,
%T A128833 34842114263551,1120413075641343,35940921946155007,
%U A128833 1151514816750309375,36870975646169341951,1180231376725002502143,37773167607267111108607
%N A128833 Number of n-tuples where each entry is chosen from the subsets of {1,2,3,4,5} such that the intersection of all n entries is empty.
%C A128833 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 A128833 Richard P. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth & Brooks, 1986,  p. 11.
%H A128833 Harvey P. Dale, <a href="/A128833/b128833.txt">Table of n, a(n) for n = 1..664</a>
%H A128833 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (63,-1302,11160,-41664,64512,-32768).
%F A128833 a(n) = (2^n-1)^5
%F A128833 G.f.: x*(1024*x^4+5760*x^3+2800*x^2+180*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)*(32*x-1)). [_Colin Barker_, Nov 17 2012]
%e A128833 a(1)=(2^1-1)^5=1 because only one tuple of length one, namely ({}) has an empty intersection of its sole entry.
%p A128833 for k from 1 to 20 do (2^k-1)^5; od;
%t A128833 (2^Range[20]-1)^5 (* or *) LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{1,243,16807,759375,28629151,992436543},20] (* or *) CoefficientList[Series[x (1024x^4+5760x^3+2800x^2+180x+1)/((x-1)(2x-1)(4x-1)(8x-1)(16x-1)(32x-1)),{x,0,20}],x] (* _Harvey P. Dale_, Aug 16 2021 *)
%Y A128833 Cf. A060867.
%K A128833 easy,nonn
%O A128833 1,2
%A A128833 Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007