A039899 Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).

0, 0, 1, 2, 3, 5, 8, 12, 18, 25, 36, 49, 68, 91, 123, 162, 214, 278, 362, 464, 596, 757, 961, 1209, 1521, 1897, 2366, 2931, 3627, 4463, 5487, 6711, 8200, 9976, 12121, 14672, 17738, 21371, 25716, 30852, 36964, 44168, 52709, 62746, 74600, 88497

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Comments

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

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

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A003106, A003114, A039900, A237976.

Programs

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 {1, 4}, 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, 24}] (* 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[{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 *)

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

Formula

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