OFFSET
0,3
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.
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)/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 *)
CROSSREFS
First differences are A027187.
The version for even instead of odd greatest part is A306145.
A000070 counts partitions with alternating sum 1.
A067661 counts strict partitions of even length.
A344610 counts partitions by sum and positive reverse-alternating sum.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 19 2018
STATUS
approved