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

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, 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, 10, 95, 250, 353, 343, 247, 129, 46, 10, 1, 0, 11, 125, 386, 628, 701, 590, 376, 175, 56, 11, 1

Offset: 0

Author

Peter Bala, Jan 19 2015

Keywords

• nonn
• easy
• tabl

Comments

A refinement of A227682.

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, 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.

T(n,k) equals the number of colored compositions of n into k parts.

See A253830 for the enumeration of colored compositions having parts with distinct colors.

Examples

			Triangle begins
n\k| 0  1   2   3   4   5   6  7
= = = = = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  2   1
3  | 0  3   3   1
4  | 0  4   7   4   1
5  | 0  5  13  11   5   1
6  | 0  6  22  25  16   6  1
7  | 0  7  34  50  41  22  7  1
...
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:
2(c1) + 2(c1), 2(c1) + 2(c2), 2(c2) + 2(c2),
1(c1) + 3(c1), 1(c1) + 3(c2), 1(c1) + 3(c3),
3(c1) + 1(c1).
		

Links

• P. Bala, Colored Compositions

Crossrefs

Cf. A008284, A227682 (row sums), A253830.

Programs

Maple

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;

Formula

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)) ) =

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 + ....

Note, G(x*(1 - z),z) is the generating function of A008284.

T(n,k) = Sum_{i = k..n} binomial(i-1,k-1)*A008284(n+k-i,k).

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.

Row sums are A227682.