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A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5). 15
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 (list; graph; refs; listen; history; text; internal format)
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

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 *)

CROSSREFS

Cf. A003106, A003114, A039899.

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

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

Sequence in context: A026906 A164315 A171861 * A039902 A081659 A143586

Adjacent sequences:  A039897 A039898 A039899 * A039901 A039902 A039903

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)