A306187 Number of n-times partitions of n.

1, 1, 3, 10, 65, 371, 3780, 33552, 472971, 5736082, 97047819, 1547576394, 32992294296, 626527881617, 15202246707840, 352290010708120, 9970739854456849, 262225912049078193, 8309425491887714632, 250946978120046026219, 8898019305511325083149

Offset

0,3

Comments

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - Gus Wiseman, Jan 27 2019

Links

Alois P. Heinz, Table of n, a(n) for n = 0..410

Wikipedia, Partition (number theory)

Formula

a(n) = A323718(n,n).

Example

From Gus Wiseman, Jan 27 2019: (Start)
The a(1) = 1 through a(3) = 10 partitions:
  (1)  ((2))     (((3)))
       ((11))    (((21)))
       ((1)(1))  (((111)))
                 (((2)(1)))
                 (((11)(1)))
                 (((2))((1)))
                 (((1)(1)(1)))
                 (((11))((1)))
                 (((1)(1))((1)))
                 (((1))((1))((1)))
(End)

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

Mathematica

ptnlevct[n_, k_]:=Switch[k, 0, 1, 1, PartitionsP[n], _, SeriesCoefficient[Product[1/(1-ptnlevct[m, k-1]*x^m), {m, n}], {x, 0, n}]];
Table[ptnlevct[n, n], {n, 0, 8}] (* Gus Wiseman, Jan 27 2019 *)

Crossrefs

Main diagonal of A323718.

Cf. A000041, A306188.

Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

Sequence in context: A167939 A352766 A206724 * A009400 A217388 A004102

Adjacent sequences: A306184 A306185 A306186 * A306188 A306189 A306190

Keyword

nonn

Author

Alois P. Heinz, Jan 27 2019

Status

approved