A346634 -id:A346634 - cp's OEIS frontend

A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k(2k+1)) / Product_{j=1..2*k} (1 - x^j).

1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 44, 55, 68, 84, 103, 126, 153, 185, 223, 268, 320, 381, 452, 535, 631, 742, 870, 1018, 1188, 1383, 1607, 1863, 2155, 2489, 2869, 3301, 3792, 4348, 4978, 5691, 6496, 7404, 8428, 9580, 10875, 12330, 13962, 15791, 17840, 20131, 22691

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

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.
- Allowing any number of odd parts gives A346634 (bisection of A067659).
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.
Index entries for sequences related to partitions

Cf. A036469, A078616, A304620, A318156.
First differences are A067661 (non-strict: A027187, odd bisection: A343942).
A000041 counts partitions.
A000070 counts partitions with alternating sum 1.
A078408 counts strict partitions of 2n+1 (odd bisection of A000009).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A027193, A035294, A067659, A087447, A236559, A236914, A239829, A306145, A344611, A344739, A346634.

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 *)

a(n) = A036469(n) - A318156(n).
a(n) = A318156(n) + A078616(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 20 2018

cp's OEIS Frontend

A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k(2k+1)) / Product_{j=1..2*k} (1 - x^j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A318155 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k(2k+1)) / Product_{j=1..2*k} (1 - x^j).