0,2

Define a unit column of a binary matrix to be a column with only one 1. a(n)=number of 3 x n binary matrices with 1 unit column up to row and column permutations(if offset is 1) - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 09 2000

Generally, number of 3 x n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is cycle index of symmetric group S_n of degree n.

First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).

Transform of tetrahedral numbers binomial(n+3,3) under Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

a(n) = A108561(n+5,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2005

Dragomir Z. Djokovic, Poincare series of some pure and mixed trace algebras of two generic matrices. See Table 4.

[((n+3)^2-1)*((n+3)^2-3)/48].

G.f.: 1/((1+x)*(1-x)^5).

Sum{k=0..n, (-1)^(n-k)C(k+4, 4) } - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

{a[n] == a[n - 2] + (n*(n + 1)*(n - 1))/6, a[1] == 0, a[2] == 1}; (3*(-1)^n - 3*(-1)^(2*n) + 12*n - 20*(-1)^(2*n)*n + 22*n^2 - 18*(-1)^(2*n)*n^2 + 12*n^3 - 4*(-1)^(2*n)*n^3 + 2*n^4)/96 - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004

a(n)=sum{k=0..floor(n/2), binomial(n-2k+3, 3)}; a(n)=sum{k=0..n, binomial(k+3, 3)*(1-(-1)^(n+k-1))/2}; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:

[0 0] [0 0] [0 1] [0 1]

[0 0] [0 1] [0 1] [0 1]

[0 1] [1 1] [1 0] [1 1].

(PARI) a(n)=if(n<0, 0, ((n+3)^2-1)*((n+3)^2-3)/48-if(n%2, 1/16))

Cf. A057524, A056885.

A002620(n)=a(n-2)-2*a(n-3)+a(n-4). A000332(n)=a(n-4)+a(n-5). A000292(n)=a(n)-a(n-2).

Sequence in context: A014818 A006527 A057304 this_sequence A007678 A115294 A110610

Adjacent sequences: A001749 A001750 A001751 this_sequence A001753 A001754 A001755

nonn

njas