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 A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 0, 6, 4, 1, 0, 1, 1, 4, 4, 5, 1, 0, 1, 0, 9, 8, 15, 6, 1, 0, 1, 1, 9, 5, 15, 21, 7, 1, 0, 1, 0, 10, 8, 20, 6, 28, 8, 1, 0, 1, 1, 12, 12, 6, 96, 42, 36, 9, 1, 0, 1, 0, 15, 12, 30, 192, 168, 64, 45, 10, 1, 0, 1, 1, 13, 9, 20, 142, 238, 204, 93, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 LINKS Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE T(5,1) = 1: [5]. T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1]. T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1]. T(5,5) = 1: [1,1,1,1,1]. Triangle T(n,k) begins:   1;   0, 1;   0, 1, 1;   0, 1, 0,  1;   0, 1, 1,  3,  1;   0, 1, 0,  6,  4,  1;   0, 1, 1,  4,  4,  5,  1;   0, 1, 0,  9,  8, 15,  6,  1;   0, 1, 1,  9,  5, 15, 21,  7,  1;   0, 1, 0, 10,  8, 20,  6, 28,  8, 1;   0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1; MAPLE b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,       `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, x^j*        b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2, {})): seq(T(n), n=0..16); MATHEMATICA b[n_, i_, s_] := b[n, i, s] = If[n==0, Total[s]!, If[i<1, 0, Expand[Sum[ If[j>0 && MemberQ[s, j], 0, x^j*b[n-i*j, i-1, If[j==0, s, s ~Union~ {j}] ]/j!], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *) PROG (PARI) T(n)={Vecrev(((r, k, b, w)->if(!k||!r, if(r, 0, w!*x^w), sum(m=0, r\k, if(!m || !bittest(b, m), self()(r-k*m, k-1, bitor(b, 1<1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780. Row sums give A242882. T(2n,n) gives A321772. Cf. A242896. Sequence in context: A152892 A193002 A122960 * A307753 A181116 A051834 Adjacent sequences:  A242884 A242885 A242886 * A242888 A242889 A242890 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 25 2014 STATUS approved

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Last modified August 7 14:42 EDT 2020. Contains 336276 sequences. (Running on oeis4.)