A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).

0, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 135, 187, 253, 343, 456, 607, 797, 1045, 1355, 1755, 2252, 2884, 3666, 4651, 5863, 7375, 9226, 11517, 14310, 17741, 21904, 26988, 33130, 40586, 49558, 60394, 73383, 88996, 107642, 129958, 156519, 188178, 225734, 270335, 323078, 385494

Offset: 0

Author

Ilya Gutkovskiy, Aug 19 2018

Keywords

• nonn

Comments

Partial sums of A027193.

From Gus Wiseman, Jun 23 2021: (Start)

Also the number of even-length integer partitions of 2n+1 with exactly one odd part. For example, the a(1) = 1 through a(5) = 10 partitions are:

(2,1) (3,2) (4,3) (5,4) (6,5)

(4,1) (5,2) (6,3) (7,4)

(6,1) (7,2) (8,3)

(2,2,2,1) (8,1) (9,2)

(3,2,2,2) (10,1)

(4,2,2,1) (4,3,2,2)

(4,4,2,1)

(5,2,2,2)

(6,2,2,1)

(2,2,2,2,2,1)

Also partitions of 2n+1 with even greatest part and alternating sum 1.

(End)

Links

• Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
• Index entries for sequences related to partitions

Crossrefs

First differences are A027193.

The ordered version appears to be A087447 modulo initial terms.

The version for odd instead of even-length partitions is A304620.

The case of strict partitions is A318156.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A027187 counts partitions of even length, with strict case A067661.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A344610 counts partitions by sum and positive reverse-alternating sum.

Cf. A000070, A000097, A006330, A030229, A067659, A236559, A236914, A239829, A239830, A338907, A344611.

Programs

Mathematica

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 23 2021 *)

Formula

a(n) = A000070(n) - A304620(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018