

A253829


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


3



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


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.


LINKS

Table of n, a(n) for n=0..77.
P. Bala, Colored Compositions


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(i1,k1)*A008284(n+ki,k).
Recurrence equation: T(n,k) = T(n1,k) + T(n1,k1) + T(nk,k)  T(nk1,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.


EXAMPLE

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).


MAPLE

G := 1/(product(1x*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. A008284, A227682 (row sums), A253830.
Sequence in context: A199011 A206735 A089000 * A107238 A258170 A055830
Adjacent sequences: A253826 A253827 A253828 * A253830 A253831 A253832


KEYWORD

nonn,easy,tabl


AUTHOR

Peter Bala, Jan 19 2015


STATUS

approved



