%I #19 Sep 25 2018 15:25:06
%S 1,1,2,1,2,3,1,3,4,5,1,3,6,7,7,1,4,8,12,12,11,1,4,10,16,21,19,15,1,5,
%T 12,23,31,36,30,22,1,5,15,28,45,55,58,45,30,1,6,17,37,60,84,94,92,67,
%U 42,1,6,20,44,80,115,147,153,140,97,56,1,7,23,55,101,161,211,249,244,211,139,77
%N Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.
%C Number of ways to factor p^(n-k)*q^k where p and q are distinct primes and each factor is a multiple of q.
%H Alois P. Heinz, <a href="/A091438/b091438.txt">Rows n = 1..141, flattened</a>
%F G.f.: A(x,y) = Product_{i>=1, j=1..i} (1/(1-x^i*y^j)).
%e 1;
%e 1, 2;
%e 1, 2, 3;
%e 1, 3, 4, 5;
%e 1, 3, 6, 7, 7; ...
%p b:= proc(n, i, j, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p `if`(i<1 or k<1, 0, `if`(j<1, b(n, i-1, i-1, k),
%p b(n, i, j-1, k)+`if`(i>n or j>k, 0, b(n-i, i, j, k-j)))))
%p end:
%p a:= (n, k)-> b(n$2, k$2):
%p seq(seq(a(n,k), k=1..n), n=1..15); # _Alois P. Heinz_, Mar 14 2015
%t b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, If[k == 0, 1, 0], If[i < 1 || k < 1, 0, If[j < 1, b[n, i - 1, i - 1, k], b[n, i, j - 1, k] + If[i > n || j > k, 0, b[n - i, i, j, k - j]]]]]; a[n_, k_] := b[n, n, k, k]; Table[a[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 10 2016, after _Alois P. Heinz_ *)
%Y Row sums: A000219.
%Y Main diagonal: A000041.
%Y a(2n,n) gives A108457.
%Y Cf. A054225.
%K nonn,tabl
%O 1,3
%A _Wouter Meeussen_ and _Christian G. Bower_, Jan 08 2004