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A331570 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and columns in decreasing lexicographic order. 11
1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 3, 1, 0, 1, 46, 42, 3, 1, 0, 1, 544, 1900, 268, 5, 1, 0, 1, 7983, 184550, 73028, 1239, 11, 1, 0, 1, 144970, 29724388, 57835569, 2448599, 7278, 13, 1, 0, 1, 3097825, 7137090958, 99940181999, 16550232235, 75497242, 40828, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.

LINKS

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

FORMULA

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331568(n, j)/k!.

A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331572(n, j).

A331708(n) = Sum_{d|n} A(n/d, d).

EXAMPLE

Array begins:

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

n\k | 0  1    2        3             4                  5

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

  0 | 1  1    0        0             0                  0 ...

  1 | 1  1    1        1             1                  1 ...

  2 | 1  1    6       46           544               7983 ...

  3 | 1  3   42     1900        184550           29724388 ...

  4 | 1  3  268    73028      57835569        99940181999 ...

  5 | 1  5 1239  2448599   16550232235    311353753947045 ...

  6 | 1 11 7278 75497242 4388476386528 896320470282357104 ...

  ...

The A(2,2) = 6 matrices are:

   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]

   [1 0]  [1 1]  [0 1]  [0 1]  [0 2]  [1 2]

   [0 1]  [0 1]  [1 1]

PROG

(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]]++); binomial(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, k<=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]))); }

CROSSREFS

Rows 1..3 are A000012, A331704, A331705.

Columns k=0..3 are A000012, A032020, A331706, A331707.

Cf. A331315, A331568, A331569, A331571, A331572, A331708.

Sequence in context: A102525 A119923 A204420 * A102410 A105123 A058291

Adjacent sequences:  A331567 A331568 A331569 * A331571 A331572 A331573

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 21 2020

STATUS

approved

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Last modified June 5 03:12 EDT 2020. Contains 334828 sequences. (Running on oeis4.)