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A331567
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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.
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9
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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
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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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A(n,k) = 0 for k > 0, n > 2^(k-1).
A(2^(k-1), k) = (2^k-1)! for k > 0.
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EXAMPLE
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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]
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PROG
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(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]))); }
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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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