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

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, 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, 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

Offset: 0

Author

Peter Bala, Jan 20 2015

Keywords

• nonn
• easy
• tabl

Comments

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. 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).

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.

Examples

			Triangle begins
n\k| 0  1  2  3  4  5  6  7
= = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  1  1
3  | 0  1  1  2
4  | 0  1  1  3  2
5  | 0  1  1  4  3  3
6  | 0  1  1  5  4  5  4
7  | 0  1  1  6  5  7  8  4
...
Row 5 polynomial: x + x^2 + 4*x^3 + 3*x*4 + 3*x^5.
Colored             x^(weight of color partition)
compositions
of 5 with
distinct colored
parts
= = = = = = = = = = = = = = = = = = = = = =
5(c1)                        x
5(c2)                        x^2
1(c1) + 4(c2)                x^3
2(c1) + 3(c2)                x^3
3(c1) + 2(c2)                x^3
5(c3)                        x^3
1(c1) + 4(c3)                x^4
2(c1) + 3(c3)                x^4
5(c4)                        x^4
1(c1) + 4(c4)                x^5
2(c2) + 3(c3)                x^5
5(c5)                        x^5
		

Links

• P. Bala, Colored Compositions

Crossrefs

Cf. A008289, A126348 (row sums), A253829.

Programs

Maple

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;

Formula

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.

T(n,k) = Sum_{i = 1..k} binomial(i+n-k-1,i-1)*A008289(k,i).

Row sums are A126348.