A182485 Number of partitions of n into exactly k different parts with distinct 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, 1, 0, 2, 3, 0, 4, 3, 0, 2, 8, 0, 4, 9, 0, 3, 12, 0, 4, 16, 1, 0, 2, 22, 4, 0, 6, 20, 5, 0, 2, 31, 12, 0, 4, 35, 16, 0, 4, 34, 24, 0, 5, 44, 33, 0, 2, 51, 52, 0, 6, 53, 57, 0, 2, 62, 89, 0, 6, 65, 100, 1, 0, 4, 68, 131, 5, 0, 4, 87
Offset: 0
Examples
T(0,0) = 1: []. T(1,1) = 1: [1]. T(2,1) = 2: [1,1], [2]. T(4,1) = 3: [1,1,1,1], [2,2], [4]. T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice. T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1]. T(7,2) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1]. T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10]. T(10,3) = 1: [3,2,2,1,1,1]. Triangle T(n,k) begins: 1; 0, 1; 0, 2; 0, 2; 0, 3, 1; 0, 2, 3; 0, 4, 3; 0, 2, 8; 0, 4, 9; 0, 3, 12; 0, 4, 16, 1;
Links
- Alois P. Heinz, Rows n = 0..120, flattened
Crossrefs
Programs
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Maple
b:= proc(n, i, t, s) option remember; `if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+ add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i)))) end: g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end: T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)): seq(T(n), n=0..30); -
Mathematica
b[n_, i_, t_, s_] := b[n, i, t, s] = If[Length[s] > t, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, t, s] + Sum[If[MemberQ[s, j], 0, b[n-i*j, i-1, t, s ~Union~ {j}]], {j, 1, n/i}]]]]; g[n_] := Module[{i}, For[ i = 1, i*(i+1)*(i+2)/6 <= n , i++]; i-1 ]; t[n_] := Table [b[n, n, k, {}] - b[n, n, k-1, {}], {k, 0, g[n]}]; Table [t[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)