A331568 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of distinct nonzero rows with column sums n.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 3, 1, 1, 75, 313, 87, 3, 1, 1, 541, 14797, 11655, 539, 5, 1, 1, 4683, 1095601, 4498191, 439779, 2483, 11, 1, 1, 47293, 119621653, 3611504823, 1390686419, 14699033, 14567, 13, 1, 1, 545835, 17943752233, 5192498314767, 12006713338683, 397293740555, 453027131, 81669, 19, 1

Offset

0,8

Links

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

Formula

A331648(n) = Sum_{d|n} A(n/d, d).

Example

Array begins:
================================================================
n\k | 0  1     2         3               4                 5
----+-----------------------------------------------------------
  0 | 1  1     1         1               1                 1 ...
  1 | 1  1     3        13              75               541 ...
  2 | 1  1    13       313           14797           1095601 ...
  3 | 1  3    87     11655         4498191        3611504823 ...
  4 | 1  3   539    439779      1390686419    12006713338683 ...
  5 | 1  5  2483  14699033    397293740555 37366422896708825 ...
  6 | 1 11 14567 453027131 105326151279287 ...
  ...
The A(2,2) = 13 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]
.
   [2 1]  [2 0]  [1 2]  [1 0]  [0 2]  [0 1]  [2 2]
   [0 1]  [0 2]  [1 0]  [1 2]  [2 0]  [2 1]

Prog

(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(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=0..3 are A000012, A000670, A331644, A331645.

Columns k=0..3 are A000012, A032020, A331646, A331647.

Cf. A219585, A331315, A331567, A331570, A331572, A331648.

Sequence in context: A257565 A276121 A262809 * A010278 A137795 A265025

Adjacent sequences: A331565 A331566 A331567 * A331569 A331570 A331571

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 21 2020

Status

approved