A331567 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column.

1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 13, 6, 0, 1, 1, 75, 120, 0, 0, 1, 1, 541, 6174, 1104, 0, 0, 1, 1, 4683, 449520, 413088, 5040, 0, 0, 1, 1, 47293, 49686726, 329520720, 18481080, 0, 0, 0, 1, 1, 545835, 7455901320, 491236986720, 179438982360, 522481680, 0, 0, 0, 1

Offset

0,8

Links

Andrew Howroyd, Table of n, a(n) for n = 0..209

Formula

A(n,k) = 0 for k > 0, n > 2^(k-1).

A(2^(k-1), k) = (2^k-1)! for k > 0.

A331643(n) = Sum_{d|n} A(n/d, d).

Example

Array begins:
===============================================================
n\k | 0 1 2    3          4              5                6
----+----------------------------------------------------------
  0 | 1 1 1    1          1              1                1 ...
  1 | 1 1 3   13         75            541             4683 ...
  2 | 1 0 6  120       6174         449520         49686726 ...
  3 | 1 0 0 1104     413088      329520720     491236986720 ...
  4 | 1 0 0 5040   18481080   179438982360 3785623968170400 ...
  5 | 1 0 0    0  522481680 70302503250720 ...
  6 | 1 0 0    0 7875584640 ...
  ...
The A(2,2) = 6 matrices are:
   [1 1]  [1 1]  [1 0]  [1 0]  [0 1]  [0 1]
   [1 0]  [0 1]  [1 1]  [0 1]  [1 1]  [1 0]
   [0 1]  [1 0]  [0 1]  [1 1]  [1 0]  [1 1]

Prog

(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Crossrefs

Rows n=1..3 are A000670, A331640, A331641.

Column k=5 is A331642.

Cf. A188445, A330942, A331568, A331569, A331571, A331643.

Sequence in context: A126308 A370040 A094923 * A303301 A160499 A329272

Adjacent sequences: A331564 A331565 A331566 * A331568 A331569 A331570

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 20 2020

Status

approved