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A331568
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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.
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10
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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
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OFFSET
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0,8
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LINKS
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FORMULA
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EXAMPLE
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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]
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PROG
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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]))); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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