A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

1, 1, 3, 3, 7, 3, 12, 3, 16, 8, 14, 3, 39, 3, 16, 15, 40, 3, 50, 3, 54, 17, 20, 3, 135, 10, 22, 25, 73, 3, 129, 3, 119, 21, 26, 19, 273, 3, 28, 23, 217, 3, 203, 3, 123, 74, 32, 3, 590, 12, 106, 27, 154, 3, 370, 23, 343, 29, 38, 3, 963, 3, 40, 95, 450, 25, 467, 3

Offset: 0

Author

Gus Wiseman, Jan 27 2019

Keywords

• nonn

Comments

An unlabeled version of A279789.

Examples

			The a(1) = 1 through a(6) = 12 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (111)      (22)          (11111)          (33)
       (1)(1)  (1)(1)(1)  (1111)        (1)(1)(1)(1)(1)  (222)
                          (2)(2)                         (3)(3)
                          (2)(11)                        (111111)
                          (11)(11)                       (3)(111)
                          (1)(1)(1)(1)                   (2)(2)(2)
                                                         (111)(111)
                                                         (2)(2)(11)
                                                         (2)(11)(11)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(1)(1)(1)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A001970, A006171 (constant parts), A007716, A034729, A047966 (uniform partitions), A047968, A279787, A279789 (twice-partitions version), A305551 (equal part-sums), A306017, A319056, A323766, A323775, A323776.

Programs

Mathematica

Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@ptn]],{ptn,Select[IntegerPartitions[n],SameQ@@#&]}]],{n,30}]

PARI

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019

Formula

a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.