

A253830


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


2



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
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OFFSET

0,10


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.


LINKS

Table of n, a(n) for n=0..90.
P. Bala, Colored Compositions


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+nk1,i1)*A008289(k,i).
Row sums are A126348.


EXAMPLE

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


MAPLE

G := product(1+(x*z)^j/(1z), 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;


CROSSREFS

Cf. A008289, A126348 (row sums), A253829.
Sequence in context: A176076 A058725 A068446 * A167625 A107261 A265336
Adjacent sequences: A253827 A253828 A253829 * A253831 A253832 A253833


KEYWORD

nonn,easy,tabl


AUTHOR

Peter Bala, Jan 20 2015


STATUS

approved



