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A128832
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Number of n-tuples where each entry is chosen from the subsets of {1,2,3,4} such that the intersection of all n entries is empty.
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2
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1, 81, 2401, 50625, 923521, 15752961, 260144641, 4228250625, 68184176641, 1095222947841, 17557851463681, 281200199450625, 4501401006735361, 72040003462430721, 1152780773560811521, 18445618199572250625
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OFFSET
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1,2
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COMMENTS
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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}.
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REFERENCES
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Stanley, R. P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11
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LINKS
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FORMULA
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a(n) = (2^n - 1)^4.
G.f.: -x*(4*x+1)*(16*x^2+46*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). [Colin Barker, Nov 17 2012]
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EXAMPLE
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a(1) = (2^1 - 1)^4 = 1 because only one tuple of length one, namely ({}), has an empty intersection of its sole entry.
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MAPLE
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for k from 1 to 20 do (2^k-1)^4; od;
with (combinat):seq(mul(stirling2(n, 2), k=1..4), n=2..17); # Zerinvary Lajos, Dec 16 2007
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MATHEMATICA
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LinearRecurrence[{31, -310, 1240, -1984, 1024}, {1, 81, 2401, 50625, 923521}, 20] (* Harvey P. Dale, Mar 30 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
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STATUS
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approved
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