OFFSET
1,2
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}.
REFERENCES
Stanley, R. P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11
LINKS
Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).
FORMULA
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]
EXAMPLE
a(1) = (2^1 - 1)^4 = 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^k-1)^4; od;
with (combinat):seq(mul(stirling2(n, 2), k=1..4), n=2..17); # Zerinvary Lajos, Dec 16 2007
MATHEMATICA
LinearRecurrence[{31, -310, 1240, -1984, 1024}, {1, 81, 2401, 50625, 923521}, 20] (* Harvey P. Dale, Mar 30 2019 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
STATUS
approved