A329436 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).

0, 1, 1, 2, 2, 4, 3, 5, 6, 8, 7, 13, 10, 16, 18, 22, 21, 34, 29, 44, 45, 56, 56, 82, 78, 100, 109, 136, 137, 185, 181, 231, 247, 295, 317, 399, 404, 490, 533, 638, 669, 817, 853, 1020, 1108, 1276, 1371, 1638, 1728, 2017, 2186, 2519, 2702, 3153, 3371, 3885

Offset: 1

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

• nonn

Comments

Inverse Moebius transform of A025147.

Number of uniform (constant multiplicity) partitions of n not containing 1, ranked by the odd terms of A072774. - Gus Wiseman, Dec 01 2023

Examples

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:
  (2)  (3)  (4)    (5)    (6)      (7)    (8)        (9)      (10)
            (2,2)  (3,2)  (3,3)    (4,3)  (4,4)      (5,4)    (5,5)
                          (4,2)    (5,2)  (5,3)      (6,3)    (6,4)
                          (2,2,2)         (6,2)      (7,2)    (7,3)
                                          (2,2,2,2)  (3,3,3)  (8,2)
                                                     (4,3,2)  (5,3,2)
                                                              (3,3,2,2)
                                                              (2,2,2,2,2)
(End)
		

Crossrefs

The strict case is A025147.

The version allowing 1 is A047966.

The version requiring 1 is A097986.

Cf. A023645, A047968, A072774, A096765, A329435, A329436, A367586.

Programs

Mathematica

nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&SameQ@@Length/@Split[#]&]], {n,0,30}] (* Gus Wiseman, Dec 01 2023 *)

Formula

G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A025147(d).