cp's OEIS Frontend

A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)).

%I A253830 #10 Dec 13 2015 02:27:37
%S A253830 1,0,1,0,1,1,0,1,1,2,0,1,1,3,2,0,1,1,4,3,3,0,1,1,5,4,5,4,0,1,1,6,5,7,
%T A253830 8,4,0,1,1,7,6,9,13,10,6,0,1,1,8,7,11,19,16,13,8,0,1,1,9,8,13,26,23,
%U A253830 22,18,10,0,1,1,10,9,15,34,31,33,31,25,12,0,1,1,11,10,17,43,40,46,47,47,30,15
%N A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)).
%C A253830 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.
%C A253830 The color numbers thus form a partition, called the color partition, of some integer. For example, 2(c1) + 1(c1) + 5(c3) + 4(c3) + 6(c4) is a colored composition of 18 (the color number of a part is shown after the part prefaced by the letter c) and has the associated color partition (1,1,3,3,4).
%C A253830 T(n,k) equals the number of colored compositions of n whose associated color partition has distinct parts with sum (called the weight of the color partition) equal to k. An example is given below.
%H A253830 P. Bala, <a href="/A253829/a253829.pdf">Colored Compositions</a>
%F A253830 G.f.: G(x,z) := Product_{n >= 1} (1 + (x*z)^n/(1 - z)) = 1 + x*z + (x + x^2)*z^2 + (x + x^2 + 2*x^3)*z^3 + (x + x^2 + 3*x^3 + 2*x^4)*z^4 + .... Note, G(x*z/(x - 1),(x - 1)/x) is the generating function of A008289.
%F A253830 T(n,k) = Sum_{i = 1..k} binomial(i+n-k-1,i-1)*A008289(k,i).
%F A253830 Row sums are A126348.
%e A253830 Triangle begins
%e A253830 n\k| 0  1  2  3  4  5  6  7
%e A253830 = = = = = = = = = = = = = =
%e A253830 0  | 1
%e A253830 1  | 0  1
%e A253830 2  | 0  1  1
%e A253830 3  | 0  1  1  2
%e A253830 4  | 0  1  1  3  2
%e A253830 5  | 0  1  1  4  3  3
%e A253830 6  | 0  1  1  5  4  5  4
%e A253830 7  | 0  1  1  6  5  7  8  4
%e A253830 ...
%e A253830 Row 5 polynomial: x + x^2 + 4*x^3 + 3*x*4 + 3*x^5.
%e A253830 Colored             x^(weight of color partition)
%e A253830 compositions
%e A253830 of 5 with
%e A253830 distinct colored
%e A253830 parts
%e A253830 = = = = = = = = = = = = = = = = = = = = = =
%e A253830 5(c1)                        x
%e A253830 5(c2)                        x^2
%e A253830 1(c1) + 4(c2)                x^3
%e A253830 2(c1) + 3(c2)                x^3
%e A253830 3(c1) + 2(c2)                x^3
%e A253830 5(c3)                        x^3
%e A253830 1(c1) + 4(c3)                x^4
%e A253830 2(c1) + 3(c3)                x^4
%e A253830 5(c4)                        x^4
%e A253830 1(c1) + 4(c4)                x^5
%e A253830 2(c2) + 3(c3)                x^5
%e A253830 5(c5)                        x^5
%p A253830 G := 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 A253830 Cf. A008289, A126348 (row sums), A253829.
%K A253830 nonn,easy,tabl
%O A253830 0,10
%A A253830 _Peter Bala_, Jan 20 2015