A296122 Number of twice-partitions of n with no repeated partitions.

1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901

Offset

0,3

Comments

a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Example

The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2017

Mathematica

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p], UnsameQ@@#&], {p, IntegerPartitions[n]}]], {n, 15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
     Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

Sequence in context: A068034 A222082 A327287 * A293324 A284904 A084215

Adjacent sequences: A296119 A296120 A296121 * A296123 A296124 A296125

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Extensions

a(15)-a(34) from Robert G. Wilson v, Dec 06 2017

Status

approved