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

%I A319005 #27 May 11 2021 06:16:31
%S A319005 1,1,1,1,2,2,5,7,13,18,28,40,60,80,113,152,205,266,353,454,590,751,
%T A319005 959,1210,1529,1905,2381,2953,3658,4501,5539,6772,8278,10065,12230,
%U A319005 14801,17893,21544,25921,31089,37240,44478,53068,63150,75063,89018,105438,124632
%N A319005 Number of integer partitions of n whose product of parts is >= n.
%H A319005 Alois P. Heinz, <a href="/A319005/b319005.txt">Table of n, a(n) for n = 0..1000</a>
%H A319005 Pankaj Jyoti Mahanta, <a href="https://arxiv.org/abs/2010.07353">On the number of partitions of n whose product of the summands is at most n</a>, arXiv:2010.07353 [math.CO], 2020.
%e A319005 The a(1) = 1 through a(9) = 18 partitions:
%e A319005   (1)  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
%e A319005                  (22)  (32)  (33)   (43)    (44)     (54)
%e A319005                              (42)   (52)    (53)     (63)
%e A319005                              (222)  (322)   (62)     (72)
%e A319005                              (321)  (331)   (332)    (333)
%e A319005                                     (421)   (422)    (432)
%e A319005                                     (2221)  (431)    (441)
%e A319005                                             (521)    (522)
%e A319005                                             (2222)   (531)
%e A319005                                             (3221)   (621)
%e A319005                                             (3311)   (3222)
%e A319005                                             (4211)   (3321)
%e A319005                                             (22211)  (4221)
%e A319005                                                      (4311)
%e A319005                                                      (5211)
%e A319005                                                      (22221)
%e A319005                                                      (32211)
%e A319005                                                      (33111)
%p A319005 b:= proc(n, i, p) option remember; `if`(n=0 or i=1, `if`(p>1,
%p A319005       0, 1), b(n, i-1, p) +b(n-i, min(i, n-i), max(p/i, 1)))
%p A319005     end:
%p A319005 a:= n-> b(n$3):
%p A319005 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 22 2018
%t A319005 Table[Length[Select[IntegerPartitions[n],Times@@#>=n&]],{n,50}]
%t A319005 (* Second program: *)
%t A319005 b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[p > 1, 0, 1],
%t A319005      b[n, i - 1, p] + b[n - i, Min[i, n - i], Max[p/i, 1]]];
%t A319005 a[n_] := b[n, n, n];
%t A319005 a /@ Range[0, 50] (* _Jean-François Alcover_, May 11 2021, after _Alois P. Heinz_ *)
%Y A319005 Column sums of A319000.
%Y A319005 Cf. A001055, A002865, A069016, A096276, A301987, A318950, A319057, A319916.
%K A319005 nonn
%O A319005 0,5
%A A319005 _Gus Wiseman_, Oct 22 2018