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.

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

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

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}.

Examples

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

References

• Richard P. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth & Brooks, 1986, p. 11.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..664
• Index entries for linear recurrences with constant coefficients, signature (63,-1302,11160,-41664,64512,-32768).

Crossrefs

Cf. A060867.

Programs

Maple

for k from 1 to 20 do (2^k-1)^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)/((x-1)(2x-1)(4x-1)(8x-1)(16x-1)(32x-1)),{x,0,20}],x] (* Harvey P. Dale, Aug 16 2021 *)

Formula

a(n) = (2^n-1)^5

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]