OFFSET
0,4
COMMENTS
Partial sums of A067661.
From Gus Wiseman, Jul 29 2021: (Start)
Also the number of strict integer partitions of 2n+1 of odd length with exactly one odd part. For example, the a(1) = 1 through a(7) = 10 partitions are:
(1) (3) (5) (7) (9) (11) (13) (15)
(4,2,1) (4,3,2) (5,4,2) (6,4,3) (6,5,4)
(6,2,1) (6,3,2) (6,5,2) (7,6,2)
(6,4,1) (7,4,2) (8,4,3)
(8,2,1) (8,3,2) (8,5,2)
(8,4,1) (8,6,1)
(10,2,1) (9,4,2)
(10,3,2)
(10,4,1)
(12,2,1)
The following relate to these partitions:
- Not requiring odd length gives A036469.
- The non-strict version is A304620.
- The version for even instead of odd length is A318156.
(End)
LINKS
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.
FORMULA
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018
MATHEMATICA
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k + 1))/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] + QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1], UnsameQ@@#&&OddQ[Length[#]]&&Count[#, _?OddQ]==1&]], {n, 0, 15}] (* Gus Wiseman, Jul 29 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 19 2018
STATUS
approved