A258289 Number of partitions of 1, 2, 3, or more copies of n into distinct parts.

1, 1, 1, 3, 3, 7, 9, 17, 21, 43, 57, 109, 157, 301, 447, 895, 1307, 2663, 4207, 8463, 13283, 28489, 45151, 95485, 157767, 336711, 561603, 1236963, 2061173, 4567227, 7946575, 17516101, 30324977, 69519697, 121465499, 276609723, 496333307, 1137900605

Offset: 0

Author

Alois P. Heinz, May 25 2015

Keywords

• nonn

Examples

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [2].
a(3) = 3: [3], [2,1], [3;2,1].
a(4) = 3: [4], [3,1], [4;3,1].
a(5) = 7: [5], [4,1], [3,2], [5;4,1], [5;3,2], [4,1;3,2], [5;4,1;3,2].
a(7) = 17: [7], [6,1], [5,2], [4,3], [4,2,1], [7;6,1], [7;5,2], [7;4,3], [7;4,2,1], [6,1;5,2], [6,1;4,3], [5,2;4,3], [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3], [7;6,1;5,2;4,3].
		

Crossrefs

Cf. A000009, A065033, A258280.

Programs

Maple

b:= proc() option remember; local m; m:= args[nargs];
     `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),
     `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,
      b(args[t] $t=1..nargs-1, m-1)+add(`if`(args[j]-m<0, 0,
      b(sort([seq(args[i]-`if`(i=j, m, 0), i=1..nargs-1)])[]
      , m-1)), j=1..nargs-1))))
    end:
a:= n-> add(b(n$k+1)/k!, k=1..max(1, ceil(n/2))):
seq(a(n), n=0..20);

Mathematica

disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];
T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;
a[n_] := a[n] = If[n == 0, 1, Sum[T[n, k], {k, 1, Quotient[n+1, 2]}]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, May 01 2022 *)

Formula

a(n) = Sum_{k=1..A065033(n)} A258280(n,k).

a(n) = Sum_{k=1..max(1,ceiling(n/2))} 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).