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%I #12 Jan 10 2020 16:19:53
%S 1,1,1,1,2,1,1,5,2,1,1,15,8,2,1,1,52,59,8,2,1,1,203,652,109,8,2,1,1,
%T 877,9736,3623,128,8,2,1,1,4140,186478,200522,11087,128,8,2,1,1,21147,
%U 4421018,16514461,2232875,21380,128,8,2,1,1,115975,126317785,1912959395,775098224,15312665,29228,128,8,2,1
%N Array read by antidiagonals: A(n,k) is the number of sets of nonempty subsets of a k-element set where each element appears in at most n subsets.
%C A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with rows in decreasing order and at most n ones in every column.
%H Andrew Howroyd, <a href="/A330964/b330964.txt">Table of n, a(n) for n = 0..209</a>
%F Lim_{n->oo} A(n,k) = 2^k.
%e Array begins:
%e ==================================================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 2 5 15 52 203 877 4140 ...
%e 2 | 1 2 8 59 652 9736 186478 4421018 ...
%e 3 | 1 2 8 109 3623 200522 16514461 1912959395 ...
%e 4 | 1 2 8 128 11087 2232875 775098224 428188962261 ...
%e 5 | 1 2 8 128 21380 15312665 22165394234 57353442460140 ...
%e 6 | 1 2 8 128 29228 70197998 422059040480 5051078354829005 ...
%e 7 | 1 2 8 128 32297 227731312 5686426671375 ...
%e ...
%e The T(1,2) = 5 set systems are:
%e {},
%e {{1,2}},
%e {{1,2}, {2}},
%e {{1},{1,2}},
%e {{1}, {2}}.
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); (vecsum(WeighT(v)) + 1)^k/prod(i=1, #v, i^v[i]*v[i]!)}
%o T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}
%Y Rows n=0..4 are A000012, A000110, A178165, A178171, A178173.
%Y Cf. A188445, A219585, A219727.
%K nonn,tabl
%O 0,5
%A _Andrew Howroyd_, Jan 04 2020
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