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%I #34 Jan 14 2022 11:34:48
%S 0,1,1,2,4,6,9,13,19,27,38,52,71,95,127,167,220,285,370,474,607,770,
%T 976,1226,1540,1920,2391,2960,3660,4501,5529,6760,8254,10038,12190,
%U 14750,17825,21470,25825,30975,37101,44322,52879,62937,74811,88733,105110,124261
%N Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).
%C For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C Short: o < 0 + 1 + 4 (OMZAAp).
%C 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
%C The Heinz numbers of these integer partitions are given by A324561. - _Gus Wiseman_, Mar 09 2019
%C From _Gus Wiseman_, Mar 09 2019: (Start)
%C 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:
%C (1) (11) (21) (22) (32) (42) (52)
%C (111) (31) (41) (51) (61)
%C (211) (221) (222) (322)
%C (1111) (311) (321) (331)
%C (2111) (411) (421)
%C (11111) (2211) (511)
%C (3111) (2221)
%C (21111) (3211)
%C (111111) (4111)
%C (22111)
%C (31111)
%C (211111)
%C (1111111)
%C (End)
%H Alois P. Heinz, <a href="/A039900/b039900.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Sum_{k>=0} x^k * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - _Seiichi Manyama_, Jan 13 2022
%e From _Gus Wiseman_, Mar 09 2019: (Start)
%e The a(1) = 1 through a(7) = 13 integer partitions with at least one part equal to 0, 1, or 4 modulo 5:
%e (1) (11) (21) (4) (5) (6) (43)
%e (111) (31) (41) (42) (52)
%e (211) (221) (51) (61)
%e (1111) (311) (321) (331)
%e (2111) (411) (421)
%e (11111) (2211) (511)
%e (3111) (2221)
%e (21111) (3211)
%e (111111) (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%e (End)
%p b:= proc(n, i, t) option remember; `if`(n=0, t,
%p `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
%p `if`(irem(i, 5) in {2, 3}, t, 1)))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 03 2014
%t Table[Count[IntegerPartitions[n], p_ /; Min[p] <= Length[p]], {n, 40}] (* _Clark Kimberling_, Feb 13 2014 *)
%t 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_ *)
%o (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
%Y Cf. A003106, A003114, A039899.
%Y Cf. A003114, A006141, A047993, A064174, A117144.
%Y Cf. A324518, A324520, A324522, A324560, A324561.
%K nonn
%O 0,4
%A _Olivier Gérard_
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