A319005 Number of integer partitions of n whose product of parts is >= n.

1, 1, 1, 1, 2, 2, 5, 7, 13, 18, 28, 40, 60, 80, 113, 152, 205, 266, 353, 454, 590, 751, 959, 1210, 1529, 1905, 2381, 2953, 3658, 4501, 5539, 6772, 8278, 10065, 12230, 14801, 17893, 21544, 25921, 31089, 37240, 44478, 53068, 63150, 75063, 89018, 105438, 124632

Offset

0,5

Links

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

Pankaj Jyoti Mahanta, On the number of partitions of n whose product of the summands is at most n, arXiv:2010.07353 [math.CO], 2020.

Example

The a(1) = 1 through a(9) = 18 partitions:
  (1)  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                 (22)  (32)  (33)   (43)    (44)     (54)
                             (42)   (52)    (53)     (63)
                             (222)  (322)   (62)     (72)
                             (321)  (331)   (332)    (333)
                                    (421)   (422)    (432)
                                    (2221)  (431)    (441)
                                            (521)    (522)
                                            (2222)   (531)
                                            (3221)   (621)
                                            (3311)   (3222)
                                            (4211)   (3321)
                                            (22211)  (4221)
                                                     (4311)
                                                     (5211)
                                                     (22221)
                                                     (32211)
                                                     (33111)

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, `if`(p>1,
      0, 1), b(n, i-1, p) +b(n-i, min(i, n-i), max(p/i, 1)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 22 2018

Mathematica

Table[Length[Select[IntegerPartitions[n], Times@@#>=n&]], {n, 50}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[p > 1, 0, 1],
     b[n, i - 1, p] + b[n - i, Min[i, n - i], Max[p/i, 1]]];
a[n_] := b[n, n, n];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

Crossrefs

Column sums of A319000.

Cf. A001055, A002865, A069016, A096276, A301987, A318950, A319057, A319916.

Sequence in context: A265795 A095295 A095333 * A095326 A095330 A036378

Adjacent sequences: A319002 A319003 A319004 * A319006 A319007 A319008

Keyword

nonn

Author

Gus Wiseman, Oct 22 2018

Status

approved