A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).

0, 1, 1, 2, 4, 6, 9, 13, 19, 27, 38, 52, 71, 95, 127, 167, 220, 285, 370, 474, 607, 770, 976, 1226, 1540, 1920, 2391, 2960, 3660, 4501, 5529, 6760, 8254, 10038, 12190, 14750, 17825, 21470, 25825, 30975, 37101, 44322, 52879, 62937, 74811, 88733, 105110, 124261

Offset

0,4

Comments

For a given partition cn(i,n) means the number of its parts equal to i modulo n.

Short: o < 0 + 1 + 4 (OMZAAp).

Number of partitions of n such that (greatest part) >= (multiplicity of greatest part), for n >= 1. For example, a(6) counts these 9 partitions: 6, 51, 42, 411, 33, 321, 3111, 22111, 21111. See the Mathematica program at A240057 for the sequence as a count of these partitions, along with counts of related partitions. - Clark Kimberling, Apr 02 2014

The Heinz numbers of these integer partitions are given by A324561. - Gus Wiseman, Mar 09 2019

From Gus Wiseman, Mar 09 2019: (Start)

Also the number of integer partitions of n whose minimum part is less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324560. For example, the a(1) = 1 through a(7) = 13 integer partitions are:

(1) (11) (21) (22) (32) (42) (52)

(111) (31) (41) (51) (61)

(211) (221) (222) (322)

(1111) (311) (321) (331)

(2111) (411) (421)

(11111) (2211) (511)

(3111) (2221)

(21111) (3211)

(111111) (4111)

(22111)

(31111)

(211111)

(1111111)

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{k>=0} x^k * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000041(n) - A003106(n). - Vaclav Kotesovec, Oct 20 2024

Example

From Gus Wiseman, Mar 09 2019: (Start)
The a(1) = 1 through a(7) = 13 integer partitions with at least one part equal to 0, 1, or 4 modulo 5:
  (1)  (11)  (21)   (4)     (5)      (6)       (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (321)     (331)
                            (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
      `if`(irem(i, 5) in {2, 3}, t, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 03 2014

Mathematica

Table[Count[IntegerPartitions[n], p_ /; Min[p] <= Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<1, 0, b[n, i-1, t] + If[i > n, 0, b[n-i, i, If[MemberQ[{2, 3}, Mod[i, 5]], t, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

Prog

(PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^k*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

Crossrefs

Cf. A003106, A003114, A039899.

Cf. A003114, A006141, A047993, A064174, A117144.

Cf. A324518, A324520, A324522, A324560, A324561.

Sequence in context: A164315 A171861 A376876 * A039902 A081659 A143586

Adjacent sequences: A039897 A039898 A039899 * A039901 A039902 A039903

Keyword

nonn

Author

Olivier Gérard

Status

approved