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 A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)). 2

%I

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

%C 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 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 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 P. Bala, <a href="/A253829/a253829.pdf">Colored Compositions</a>

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

%F Row sums are A126348.

%e Triangle begins

%e n\k| 0 1 2 3 4 5 6 7

%e = = = = = = = = = = = = = =

%e 0 | 1

%e 1 | 0 1

%e 2 | 0 1 1

%e 3 | 0 1 1 2

%e 4 | 0 1 1 3 2

%e 5 | 0 1 1 4 3 3

%e 6 | 0 1 1 5 4 5 4

%e 7 | 0 1 1 6 5 7 8 4

%e ...

%e Row 5 polynomial: x + x^2 + 4*x^3 + 3*x*4 + 3*x^5.

%e Colored x^(weight of color partition)

%e compositions

%e of 5 with

%e distinct colored

%e parts

%e = = = = = = = = = = = = = = = = = = = = = =

%e 5(c1) x

%e 5(c2) x^2

%e 1(c1) + 4(c2) x^3

%e 2(c1) + 3(c2) x^3

%e 3(c1) + 2(c2) x^3

%e 5(c3) x^3

%e 1(c1) + 4(c3) x^4

%e 2(c1) + 3(c3) x^4

%e 5(c4) x^4

%e 1(c1) + 4(c4) x^5

%e 2(c2) + 3(c3) x^5

%e 5(c5) x^5

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

%K nonn,easy,tabl

%O 0,10

%A _Peter Bala_, Jan 20 2015

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Last modified August 10 02:26 EDT 2020. Contains 336367 sequences. (Running on oeis4.)