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A323774
Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.
5
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
0,3
COMMENTS
An unlabeled version of A279789.
FORMULA
a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.
EXAMPLE
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)
MATHEMATICA
Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#], SameQ@@#&]&/@ptn]], {ptn, Select[IntegerPartitions[n], SameQ@@#&]}]], {n, 30}]
PROG
(PARI) a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019
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.
Sequence in context: A083262 A122978 A119347 * A062402 A347405 A294015
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 27 2019
STATUS
approved