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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
OFFSET
0,14
LINKS
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<<m), w+m)/m!))))(n, n, 1, 0))}
{ for(n=0, 10, print(T(n))) } \\ Andrew Howroyd, Aug 31 2019
CROSSREFS
Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.
Row sums give A242882.
T(2n,n) gives A321772.
Cf. A242896.
Sequence in context: A193002 A366725 A122960 * A307753 A181116 A051834
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 25 2014
STATUS
approved