

A128866


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 contains exactly one element.


0



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

1,1


COMMENTS

There is the following general formula: The number T(n,k,r) of ntuples 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)^(kr). This may be shown by exhibiting a bijection to a set whose cardinality is obviously C(k,r) * (2^n  1)^(kr), namely the set of all ktuples 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 (kr) entries must be chosen from the 2^n1 proper subsets of {1,..,n}, i.e. for each of the (kr) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n  1)^(kr) 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



FORMULA

a(n) = 5*(2^n1)^4.
G.f.: 5*x*(4*x+1)*(16*x^2+46*x+1)/((x1)*(2*x1)*(4*x1)*(8*x1)*(16*x1)). [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



KEYWORD

nonn,easy


AUTHOR

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


STATUS

approved



