A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

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

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

Cf. A000005 (d(n)), A304405, A304406, A304428, A304430.

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.

Cf. A098504, A098859, A275870, A353832, A353847, A353864, A353932.

Sequence in context: A133911 A069932 A056148 * A057567 A353845 A217895

Adjacent sequences: A304439 A304440 A304441 * A304443 A304444 A304445

Keyword

nonn

Author

Seiichi Manyama, May 12 2018

Status

approved