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A304442
Number of partitions of n in which the sequence of the sum of the same summands is constant.
60
1, 1, 2, 2, 4, 2, 5, 2, 7, 3, 5, 2, 13, 2, 5, 4, 11, 2, 13, 2, 12, 4, 5, 2, 28, 3, 5, 5, 12, 2, 18, 2, 17, 4, 5, 4, 44, 2, 5, 4, 24, 2, 18, 2, 12, 10, 5, 2, 63, 3, 9, 4, 12, 2, 34, 4, 24, 4, 5, 2, 67, 2, 5, 10, 27, 4, 18, 2, 12, 4, 14, 2, 120, 2, 5, 7, 12, 4, 18, 2, 54
OFFSET
0,3
COMMENTS
Said differently, these are partitions whose run-sums are all equal. - Gus Wiseman, Jun 25 2022
LINKS
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
All parts are divisors of n, see A018818, compositions A100346.
For run-lengths instead of run-sums we have A047966, compositions A329738.
These partitions are ranked by A353833.
The distinct instead of equal version is A353837, ranked by A353838, compositions A353850.
The version for compositions is A353851, ranked by A353848.
Sequence in context: A133911 A069932 A056148 * A057567 A353845 A217895
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 12 2018
STATUS
approved