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A091438 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.

%I A091438 #19 Sep 25 2018 15:25:06
%S A091438 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 A091438 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 A091438 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 A091438 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 A091438 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 A091438 Alois P. Heinz, <a href="/A091438/b091438.txt">Rows n = 1..141, flattened</a>
%F A091438 G.f.: A(x,y) = Product_{i>=1, j=1..i} (1/(1-x^i*y^j)).
%e A091438   1;
%e A091438   1, 2;
%e A091438   1, 2, 3;
%e A091438   1, 3, 4, 5;
%e A091438   1, 3, 6, 7, 7; ...
%p A091438 b:= proc(n, i, j, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A091438       `if`(i<1 or k<1, 0, `if`(j<1, b(n, i-1, i-1, k),
%p A091438        b(n, i, j-1, k)+`if`(i>n or j>k, 0, b(n-i, i, j, k-j)))))
%p A091438     end:
%p A091438 a:= (n, k)->  b(n$2, k$2):
%p A091438 seq(seq(a(n,k), k=1..n), n=1..15);  # _Alois P. Heinz_, Mar 14 2015
%t A091438 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 A091438 Row sums: A000219.
%Y A091438 Main diagonal: A000041.
%Y A091438 a(2n,n) gives A108457.
%Y A091438 Cf. A054225.
%K A091438 nonn,tabl
%O A091438 1,3
%A A091438 _Wouter Meeussen_ and _Christian G. Bower_, Jan 08 2004