A333901 Array read by antidiagonals: T(n,k) is the number of n X k nonnegative integer matrices with all column sums n and row sums k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 55, 19, 1, 1, 1, 1, 51, 415, 415, 51, 1, 1, 1, 1, 141, 3391, 10147, 3391, 141, 1, 1, 1, 1, 393, 28681, 261331, 261331, 28681, 393, 1, 1, 1, 1, 1107, 248137, 7100821, 22069251, 7100821, 248137, 1107, 1, 1

Offset

0,13

Comments

T(n,k) is the number of ordered factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..405 (antidiagonals n=0..27)

Formula

T(n,k) = T(k,n).

Example

Array begins:
=======================================================
n\k | 0 1   2     3       4          5            6
----+--------------------------------------------------
  0 | 1 1   1     1       1          1            1 ...
  1 | 1 1   1     1       1          1            1 ...
  2 | 1 1   3     7      19         51          141 ...
  3 | 1 1   7    55     415       3391        28681 ...
  4 | 1 1  19   415   10147     261331      7100821 ...
  5 | 1 1  51  3391  261331   22069251   1985311701 ...
  6 | 1 1 141 28681 7100821 1985311701 602351808741 ...
  ...
The T(3,2) = 7 matrices are:
  [1 1]  [1 1]  [1 1]  [2 0]  [2 0]  [0 2]  [0 2]
  [1 1]  [2 0]  [0 2]  [1 1]  [0 2]  [1 1]  [2 0]
  [1 1]  [0 2]  [2 0]  [0 2]  [1 1]  [2 0]  [1 1]

Prog

(PARI)
T(n, k)={
  local(M=Map(Mat([k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
}
for(n=0, 7, for(k=0, 7, print1(T(n, k), ", ")); print)

Crossrefs

Columns k=0..9 are A000012, A000012, A002426, A172743, A172816, A172868, A172904, A172931, A172947, A172961.

Main diagonal is A110058.

Cf. A257462, A257493.

Sequence in context: A147989 A119329 A334549 * A054724 A360440 A349350

Adjacent sequences: A333898 A333899 A333900 * A333902 A333903 A333904

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 18 2020

Status

approved