A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).

1, 1, 2, 3, 6, 9, 15, 22, 34, 48, 70, 97, 137, 186, 255, 341, 459, 605, 800, 1042, 1359, 1751, 2256, 2879, 3672, 4645, 5869, 7367, 9234, 11508, 14319, 17730, 21916, 26975, 33143, 40570, 49575, 60376, 73402, 88974, 107666, 129933, 156546, 188148, 225767, 270300, 323115, 385453

Offset: 0

Author

Ilya Gutkovskiy, Aug 19 2018

Keywords

• nonn

Comments

Partial sums of A027187.

From Gus Wiseman, Jun 26 2021: (Start)

Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:

1 111 32 331 54 551 76

11111 3211 3222 3332 5422

1111111 3321 5411 5521

33111 33221 33331

321111 322211 55111

111111111 332111 322222

3311111 332221

32111111 333211

11111111111 541111

3322111

32221111

33211111

331111111

3211111111

1111111111111

Also odd-length partitions of 2n+1 with exactly one odd part.

(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 A027187.

The version for even instead of odd greatest part is A306145.

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

A000070 counts partitions with alternating sum 1.

A067661 counts strict partitions of even length.

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

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

Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, A338907, A344611.

Programs

Mathematica

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {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],OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 26 2021 *)

Formula

a(n) = A000070(n) - A306145(n).

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