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A329436
Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).
3
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
1,4
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
FORMULA
G.f.: Sum_{k>=1} A025147(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A025147(d).
EXAMPLE
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)
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 *)
CROSSREFS
The strict case is A025147.
The version allowing 1 is A047966.
The version requiring 1 is A097986.
Sequence in context: A094051 A159268 A058723 * A182577 A357189 A241450
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 13 2019
STATUS
approved