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A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals. 9
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 9, 3, 1, 1, 15, 66, 31, 5, 1, 1, 52, 712, 686, 109, 7, 1, 1, 203, 10457, 27036, 6721, 339, 11, 1, 1, 877, 198091, 1688360, 911838, 58616, 1043, 15, 1, 1, 4140, 4659138, 154703688, 231575143, 26908756, 476781, 2998, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes.  A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - Andrew Howroyd, Dec 10 2018

LINKS

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

EXAMPLE

A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].

A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].

Square array A(n,k) begins:

  1,   1,    1,      1,        1,         1,         1,       1, ...

  1,   1,    2,      5,       15,        52,       203,     877, ...

  1,   2,    9,     66,      712,     10457,    198091, 4659138, ...

  1,   3,   31,    686,    27036,   1688360, 154703688, ...

  1,   5,  109,   6721,   911838, 231575143, ...

  1,   7,  339,  58616, 26908756, ...

  1,  11, 1043, 476781, ...

  1,  15, 2998, ...

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]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}

T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 11 2018

CROSSREFS

Columns k=0..3 give: A000012, A000041, A002774, A219678.

Rows n=0..3 give: A000012, A000110, A020555, A322487.

Main diagonal gives A322488.

Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.

Sequence in context: A129104 A232648 A295690 * A177694 A092450 A279629

Adjacent sequences:  A219724 A219725 A219726 * A219728 A219729 A219730

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Nov 26 2012

STATUS

approved

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Last modified September 20 16:48 EDT 2019. Contains 327242 sequences. (Running on oeis4.)