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

Author

Seiichi Manyama, May 12 2018

Keywords

• nonn

Comments

Said differently, these are partitions whose run-sums are all equal. - Gus Wiseman, Jun 25 2022

Examples

			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
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

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.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}] (* Gus Wiseman, Jun 25 2022 *)

PARI

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ Michel Marcus, May 13 2018

Formula

a(n) >= 2 for n > 1.

a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.