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Expansion of Sum_{k>=1} (-1 + Product_{j>=2} (1 + x^(k*j))).
3

%I #8 Dec 01 2023 23:26:24

%S 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,

%T 109,136,137,185,181,231,247,295,317,399,404,490,533,638,669,817,853,

%U 1020,1108,1276,1371,1638,1728,2017,2186,2519,2702,3153,3371,3885

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

%C Inverse Moebius transform of A025147.

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

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

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

%e From _Gus Wiseman_, Dec 01 2023: (Start)

%e The a(2) = 1 through a(10) = 8 uniform partitions not containing 1:

%e (2) (3) (4) (5) (6) (7) (8) (9) (10)

%e (2,2) (3,2) (3,3) (4,3) (4,4) (5,4) (5,5)

%e (4,2) (5,2) (5,3) (6,3) (6,4)

%e (2,2,2) (6,2) (7,2) (7,3)

%e (2,2,2,2) (3,3,3) (8,2)

%e (4,3,2) (5,3,2)

%e (3,3,2,2)

%e (2,2,2,2,2)

%e (End)

%t nmax = 56; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&SameQ@@Length/@Split[#]&]], {n,0,30}] (* _Gus Wiseman_, Dec 01 2023 *)

%Y The strict case is A025147.

%Y The version allowing 1 is A047966.

%Y The version requiring 1 is A097986.

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

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Nov 13 2019