

A128833


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


2



1, 243, 16807, 759375, 28629151, 992436543, 33038369407, 1078203909375, 34842114263551, 1120413075641343, 35940921946155007, 1151514816750309375, 36870975646169341951, 1180231376725002502143, 37773167607267111108607
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OFFSET

1,2


COMMENTS

The general formula where each entry is chosen from the subsets of {1,..,k} is (2^n1)^k. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n1)^k, namely the set of all ktuples with each entry chosen from the 2^n1 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}.


REFERENCES

Stanley, R.P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11


LINKS



FORMULA

a(n)=(2^n1)^5
G.f.: x*(1024*x^4+5760*x^3+2800*x^2+180*x+1)/((x1)*(2*x1)*(4*x1)*(8*x1)*(16*x1)*(32*x1)). [Colin Barker, Nov 17 2012]


EXAMPLE

a(1)=(2^11)^5=1 because only one tuple of length one, namely ({}) has an empty intersection of its sole entry.


MAPLE

for k from 1 to 20 do (2^k1)^5; od;


MATHEMATICA

(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)/((x1)(2x1)(4x1)(8x1)(16x1)(32x1)), {x, 0, 20}], x] (* Harvey P. Dale, Aug 16 2021 *)


CROSSREFS



KEYWORD

easy,nonn


AUTHOR

Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007


STATUS

approved



