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

Author

Andrew Howroyd, Apr 18 2020

Keywords

• nonn
• tabl

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).

Examples

			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]
		

Links

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

Crossrefs

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

Main diagonal is A110058.

Cf. A257462, A257493.

Programs

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)

Formula

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