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A330964 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. 6
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, 877, 9736, 3623, 128, 8, 2, 1, 1, 4140, 186478, 200522, 11087, 128, 8, 2, 1, 1, 21147, 4421018, 16514461, 2232875, 21380, 128, 8, 2, 1, 1, 115975, 126317785, 1912959395, 775098224, 15312665, 29228, 128, 8, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..209

FORMULA

Lim_{n->oo} A(n,k) = 2^k.

EXAMPLE

Array begins:

==================================================================

n\k | 0 1 2   3     4         5             6                7

----+-------------------------------------------------------------

  0 | 1 1 1   1     1         1             1                1 ...

  1 | 1 2 5  15    52       203           877             4140 ...

  2 | 1 2 8  59   652      9736        186478          4421018 ...

  3 | 1 2 8 109  3623    200522      16514461       1912959395 ...

  4 | 1 2 8 128 11087   2232875     775098224     428188962261 ...

  5 | 1 2 8 128 21380  15312665   22165394234   57353442460140 ...

  6 | 1 2 8 128 29228  70197998  422059040480 5051078354829005 ...

  7 | 1 2 8 128 32297 227731312 5686426671375 ...

      ...

The T(1,2) = 5 set systems are:

  {},

  {{1,2}},

  {{1,2}, {2}},

  {{1},{1,2}},

  {{1}, {2}}.

PROG

(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]]++); (vecsum(WeighT(v)) + 1)^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)))/(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)}

CROSSREFS

Rows n=0..4 are A000012, A000110, A178165, A178171, A178173.

Cf. A188445, A219585, A219727.

Sequence in context: A106270 A319171 A047888 * A128704 A075259 A307877

Adjacent sequences:  A330961 A330962 A330963 * A330965 A330966 A330967

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 04 2020

STATUS

approved

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Last modified September 19 17:46 EDT 2021. Contains 347564 sequences. (Running on oeis4.)