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 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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+n-k-1,i-1)*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/(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; 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

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Last modified July 8 01:48 EDT 2020. Contains 335502 sequences. (Running on oeis4.)