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Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).
7

%I #25 Jan 14 2022 11:34:33

%S 0,0,1,2,3,5,8,12,18,25,36,49,68,91,123,162,214,278,362,464,596,757,

%T 961,1209,1521,1897,2366,2931,3627,4463,5487,6711,8200,9976,12121,

%U 14672,17738,21371,25716,30852,36964,44168,52709,62746,74600,88497

%N Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).

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

%C Short: o < 0 + 2 + 3 (OMZBBp).

%C Number of partitions of n such that (greatest part) > (multiplicity of greatest part), for n >= 1. For example, a(6) counts these 8 partitions: 6, 51, 42, 411, 33, 321, 3111, 21111. See the Mathematica program at A240057 for the sequence as a count of partitions defined in this manner, and related sequences. - _Clark Kimberling_, Apr 02 2014

%H Alois P. Heinz, <a href="/A039899/b039899.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{k>=0} x^k * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - _Seiichi Manyama_, Jan 13 2022

%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 {1, 4}, 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, 24}] (* _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[{1, 4}, 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, 0], Vec(sum(k=0, N, x^k*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ _Seiichi Manyama_, Jan 13 2022

%Y Cf. A003106, A003114, A039900, A237976.

%K nonn

%O 0,4

%A _Olivier Gérard_