A242896 Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.
1, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 10, 0, 4, 12, 0, 2, 38, 0, 4, 56, 0, 3, 79, 0, 4, 152, 60, 0, 2, 251, 285, 0, 6, 284, 498, 0, 2, 594, 1438, 0, 4, 920, 2816, 0, 4, 1108, 5208, 0, 5, 2136, 11195, 0, 2, 3402, 24094, 0, 6, 4407, 38523, 0, 2, 8350, 85182
Offset: 0
Examples
T(5,1) = 2: [1,1,1,1,1], [5]. T(5,2) = 10: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1]. Triangle T(n,k) begins: 1; 0, 1; 0, 2; 0, 2; 0, 3, 3; 0, 2, 10; 0, 4, 12; 0, 2, 38; 0, 4, 56; 0, 3, 79; 0, 4, 152, 60;
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
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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, `if`(j=0, 1, x)* 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_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*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 11 2015, after Alois P. Heinz *)