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
Examples
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]
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..209
Crossrefs
Programs
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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]))); }
Formula
A331648(n) = Sum_{d|n} A(n/d, d).