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
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
KEYWORD
nonn
AUTHOR
STATUS
approved