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

0,4

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.

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

(End)

Links

Table of n, a(n) for n=0..53.

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.2 "Partitions into distinct parts", page 350.

Index entries for sequences related to partitions

Formula

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

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

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.

Sequence in context: A253170 A337567 A177332 * A282569 A213213 A319470

Adjacent sequences: A318152 A318153 A318154 * A318156 A318157 A318158

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 19 2018

Status

approved