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A253829 Triangular array with g.f. Product_{n >= 1} 1/(1 - x*z^n/(1 - z)).

%I A253829 #12 Dec 13 2015 01:35:56
%S A253829 1,0,1,0,2,1,0,3,3,1,0,4,7,4,1,0,5,13,11,5,1,0,6,22,25,16,6,1,0,7,34,
%T A253829 50,41,22,7,1,0,8,50,91,92,63,29,8,1,0,9,70,155,187,155,92,37,9,1,0,
%U A253829 10,95,250,353,343,247,129,46,10,1,0,11,125,386,628,701,590,376,175,56,11,1
%N A253829 Triangular array with g.f. Product_{n >= 1} 1/(1 - x*z^n/(1 - z)).
%C A253829 A refinement of A227682.
%C A253829 A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition. The color numbers thus form a partition, called the color partition, of some integer.
%C A253829 For example, the composition 1 + 3 + 2 of 6 gives rise to three colored compositions of 6, namely, 1(c1) + 3(c1) + 2(c1), 1(c1) + 3(c1) + 2(c2) and 1(c1) + 3(c2) + 2(c2), where the color number of a part is shown after the part prefaced by the letter c.
%C A253829 T(n,k) equals the number of colored compositions of n into k parts.
%C A253829 See A253830 for the enumeration of colored compositions having parts with distinct colors.
%H A253829 P. Bala, <a href="/A253829/a253829.pdf">Colored Compositions</a>
%F A253829 G.f.: G(x,z) := Product_{n >= 1} (1 - z)/(1 - z - x*z^n) = exp( Sum_{n >= 1} (x*z)^n/(n*(1 - z)^n*(1 - z^n)) ) =
%F A253829 1 + Sum_{n >= 1} (x*z/(1 - z))^n/( Product_{i = 1..n} 1 - z^i ) = 1 + x*z + (2*x + x^2)*z^2 + (3*x + 3*x^2 + x^3)*z^3 + ....
%F A253829 Note, G(x*(1 - z),z) is the generating function of A008284.
%F A253829 T(n,k) = Sum_{i = k..n} binomial(i-1,k-1)*A008284(n+k-i,k).
%F A253829 Recurrence equation: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-k,k) - T(n-k-1,k) with boundary conditions T(n,n) = 1, T(n,0) = 0 for n >= 1 and T(n,k) = 0 for n < k.
%F A253829 Row sums are A227682.
%e A253829 Triangle begins
%e A253829 n\k| 0  1   2   3   4   5   6  7
%e A253829 = = = = = = = = = = = = = = = = =
%e A253829 0  | 1
%e A253829 1  | 0  1
%e A253829 2  | 0  2   1
%e A253829 3  | 0  3   3   1
%e A253829 4  | 0  4   7   4   1
%e A253829 5  | 0  5  13  11   5   1
%e A253829 6  | 0  6  22  25  16   6  1
%e A253829 7  | 0  7  34  50  41  22  7  1
%e A253829 ...
%e A253829 T(4,2) = 7: The compositions of 4 into two parts are 2 + 2, 1 + 3 and 3 + 1. Coloring the parts as described above produces seven colored compositions of 4 into two parts:
%e A253829 2(c1) + 2(c1), 2(c1) + 2(c2), 2(c2) + 2(c2),
%e A253829 1(c1) + 3(c1), 1(c1) + 3(c2), 1(c1) + 3(c3),
%e A253829 3(c1) + 1(c1).
%p A253829 G := 1/(product(1-x*z^j/(1-z), j = 1 .. 12)): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;
%Y A253829 Cf. A008284, A227682 (row sums), A253830.
%K A253829 nonn,easy,tabl
%O A253829 0,5
%A A253829 _Peter Bala_, Jan 19 2015