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%I #9 Sep 22 2016 08:20:51
%S 1,1,2,1,3,3,1,5,6,4,1,7,13,10,5,1,11,22,26,15,6,1,13,40,51,45,21,7,1,
%T 19,55,103,100,71,28,8,1,23,88,161,221,176,105,36,9,1,29,118,277,386,
%U 422,287,148,45,10
%N Triangle a(n,k) = number of colors that can be produced by n units of paint from k primary colors.
%F All partitions of size n: if GCD is not 1, skip; else: fill the partition with zeros to get k numbers; count occurrences of each number (e.g.: 2 2 1 0 0 0 becomes 2 1 3); compute multinomial of k over these digits (e.g. 2 1 3 becomes 6!/(2!*1!*3!) = 60; sum.
%e Table read by antidiagonals: (1), (1 2), (1 3 3), (1 5 6 4), (1 7 13 10 5), ...
%e a(2,3)=6 because you can take each color once, or mix two colors.
%t max = 10; col[k_] := Accumulate[ Table[ Sum[ MoebiusMu[n/d]*Product[d+j, {j, 1, k}]/k!, {d, Divisors[n]}], {n, 1, max}]]; t = Table[col[k], {k, 0, max-1}] // Transpose; Flatten[ Table[ t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* _Jean-François Alcover_, Dec 26 2012 *)
%Y Row 2 = A005728, row 3 = A048134. Cf. A048240, A048241.
%K easy,nice,nonn,tabl
%O 1,3
%A _Jurjen N.E. Bos_
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