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A318156
Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).
3
0, 1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 55, 69, 85, 104, 127, 154, 186, 224, 268, 320, 381, 452, 534, 630, 741, 869, 1017, 1187, 1382, 1606, 1862, 2155, 2489, 2869, 3301, 3792, 4349, 4979, 5692, 6497, 7405, 8429, 9581, 10876, 12331, 13963, 15792, 17840, 20131, 22691
OFFSET
0,3
COMMENTS
Partial sums of A067659.
LINKS
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.
FORMULA
a(n) = A036469(n) - A318155(n).
a(n) = A318155(n) - A078616(n).
a(n) ~ exp(Pi*sqrt(n/3)) * 3^(1/4) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018
EXAMPLE
From Gus Wiseman, Jul 18 2021: (Start)
Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
(2,1) (3,2) (4,3) (5,4) (6,5) (7,6) (8,7) (9,8)
(4,1) (5,2) (6,3) (7,4) (8,5) (9,6) (10,7)
(6,1) (7,2) (8,3) (9,4) (10,5) (11,6)
(8,1) (9,2) (10,3) (11,4) (12,5)
(10,1) (11,2) (12,3) (13,4)
(12,1) (13,2) (14,3)
(6,4,2,1) (14,1) (15,2)
(6,4,3,2) (16,1)
(8,4,2,1) (6,5,4,2)
(8,4,3,2)
(8,6,2,1)
(10,4,2,1)
Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);
MATHEMATICA
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, 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@@#&&EvenQ[Length[#]]&&Count[#, _?OddQ]==1&]], {n, 0, 15}] (* Gus Wiseman, Jul 18 2021 *)
CROSSREFS
Partial sums of A067659.
The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
- Allowing any length gives A036469.
- The non-strict version is A306145.
- The version for odd length is A318155 (non-strict: A304620).
- Allowing any number of odd parts gives A343942 (odd bisection of A067661).
A000041 counts partitions.
A027187 counts partitions of even length (strict: A067661).
A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
Sequence in context: A272691 A192433 A189083 * A272948 A164001 A117598
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 19 2018
STATUS
approved