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
Main diagonal gives A322488.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 26 2012
STATUS
approved