OFFSET
0,2
COMMENTS
Unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 3 points of that set uniquely (if offset is 3).
LINKS
Author?, Table of n, a(n) for n = 0..1374
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).
FORMULA
(1/6)*(Z(S_n; 5, 5, ...)+3*Z(S_n; 3, 5, 3, 5, ...)+2*Z(S_n; 2, 2, 5, 2, 2, 5, ...)) where Z(S_n; x_1, x_2, x_3, ...) is cycle index of symmetric group S_n of degree n.
G.f.: 1/(1-x^3)/(1-x^2)/(1-x)^3.
Let P(i,k) be the number of integer partitions of n into k parts, then with k=3 we have a(n) = Sum_{m=1..n} Sum_{i=k..m} P(i,k). - Thomas Wieder, Feb 18 2007
a(n) = Sum_{m=0..n} (n-m+1)*floor(((m+3)^2+3)/12). [Renzo Benedetti, Sep 30 2009]
a(n) = floor( ((n+2)*(n+6)/12)^2 ) = round( ((n+2)*(n+6)/12)^2 ). [Renzo Benedetti, Jul 25 2012]
Partial sums of A000601. - R. J. Mathar, Jul 25 2012
EXAMPLE
There are 7 binary 3x2 matrices without unit columns up to row and column permutations:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 1] [1 1] [1 1].
MATHEMATICA
CoefficientList[ Series[ 1/(1 - x^3)/(1 - x^2)/(1 - x)^3, {x, 0, 42}], x] (* Jean-François Alcover, Mar 26 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Sep 02 2000
EXTENSIONS
More terms from James A. Sellers, Sep 07 2000
STATUS
approved