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URL: https://oeis.org/A331568

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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.
10
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
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.
Sequence in context: A257565 A276121 A262809 * A010278 A137795 A265025
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 21 2020
STATUS
approved