OFFSET
0,3
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.
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018
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 *)
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.
A344610 counts partitions by sum and positive reverse-alternating sum.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 19 2018
STATUS
approved