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 + 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
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved