A128866 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 contains exactly one element.

5, 405, 12005, 253125, 4617605, 78764805, 1300723205, 21141253125, 340920883205, 5476114739205, 87789257318405, 1406000997253125, 22507005033676805, 360200017312153605, 5763903867804057605

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..15.

Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Formula

a(n) = 5*(2^n-1)^4.

G.f.: -5*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]

Example

a(1)=5 because the five tuples of length one are ({1}),({2}),({3}),({4}),({5}).

Mathematica

LinearRecurrence[{31, -310, 1240, -1984, 1024}, {5, 405, 12005, 253125, 4617605}, 20] (* Harvey P. Dale, Nov 01 2019 *)

Prog

(Java) import java.io.*; import java.math.*; public class MakeSequence { public static void main(String[] args) { String s = new String(); BigInteger x; BigInteger one = new BigInteger("1"); BigInteger five = new BigInteger("5"); String help; try { BufferedWriter out = new BufferedWriter(new FileWriter("sequence.txt")); for (Integer k=1; k<31; ++k) { x = (((two.pow(k)).subtract(one)).pow(4)).multiply(five); help = x.toString(); s = help + ", "; out.write(s); } out.close(); } catch (IOException e) { } } }

Crossrefs

Sequence in context: A189307 A198538 A198535 * A221047 A368021 A075769

Adjacent sequences: A128863 A128864 A128865 * A128867 A128868 A128869

Keyword

nonn,easy

Author

Peter C. Heinig (heinig(AT)in.tum.de), Apr 17 2007

Status

approved