A323766 Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.

1, 1, 4, 5, 12, 9, 25, 17, 42, 39, 64, 58, 132, 103, 173, 200, 303, 299, 491, 492, 756, 832, 1122, 1257, 1858, 1975, 2646, 3083, 4057, 4567, 6118, 6844, 8913, 10265, 12912, 14931, 19089, 21639, 27003, 31397, 38830, 44585, 55138, 63263, 77371, 89585, 108076

Offset

0,3

Comments

Also the number of constant multiset partitions of constant multiset partitions of integer partitions of n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

Example

The a(6) = 25 constant multiset partitions of constant multiset partitions of integer partitions of 6:
  ((6))
  ((52))
  ((42))
  ((33))
  ((3)(3))
  ((3))((3))
  ((411))
  ((321))
  ((222))
  ((2)(2)(2))
  ((2))((2))((2))
  ((3111))
  ((2211))
  ((21)(21))
  ((21))((21))
  ((21111))
  ((111111))
  ((111)(111))
  ((11)(11)(11))
  ((111))((111))
  ((11))((11))((11))
  ((1)(1)(1)(1)(1)(1))
  ((1)(1)(1))((1)(1)(1))
  ((1)(1))((1)(1))((1)(1))
  ((1))((1))((1))((1))((1))((1))

Mathematica

Table[If[n==0, 1, Sum[PartitionsP[d]*DivisorSigma[0, n/d], {d, Divisors[n]}]], {n, 0, 30}]

Prog

(PARI) a(n) = if (n==0, 1, sumdiv(n, d, numbpart(d)*numdiv(n/d))); \\ Michel Marcus, Jan 28 2019

Crossrefs

Cf. A000005, A000041, A000837, A001970, A034729, A047968, A306017, A319066, A323764, A323765, A323774.

Sequence in context: A246316 A344372 A034773 * A330223 A325688 A080277

Adjacent sequences: A323763 A323764 A323765 * A323767 A323768 A323769

Keyword

nonn

Author

Gus Wiseman, Jan 27 2019

Status

approved