OFFSET
0,3
COMMENTS
Said differently, these are partitions whose run-sums are all equal. - Gus Wiseman, Jun 25 2022
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
FORMULA
a(n) >= 2 for n > 1.
a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.
EXAMPLE
a(72) = binomial(d(72),1) + binomial(d(36),2) + binomial(d(24),3) + binomial(d(18),4) + binomial(d(12),6) = 12 + 36 + 56 + 15 + 1 = 120, where d(n) is the number of divisors of n.
--+----------------------+-----------------------------------------
n | | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1 | 1
2 | 2 | 2
| 1+1 | 2
3 | 3 | 3
| 1+1+1 | 3
4 | 4 | 4
| 2+2 | 4
| 2+1+1 | 2, 2
| 1+1+1+1 | 4
5 | 5 | 5
| 1+1+1+1+1 | 5
6 | 6 | 6
| 3+3 | 6
| 3+1+1+1 | 3, 3
| 2+2+2 | 6
| 1+1+1+1+1+1 | 6
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], SameQ@@Total/@Split[#]&]], {n, 0, 15}] (* Gus Wiseman, Jun 25 2022 *)
PROG
(PARI) a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ Michel Marcus, May 13 2018
CROSSREFS
These partitions are ranked by A353833.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 12 2018
STATUS
approved