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A340623
The number of partitions of n without repeated odd parts having more even parts than odd parts.
4
0, 0, 1, 0, 2, 1, 3, 3, 5, 7, 8, 13, 14, 23, 23, 37, 39, 59, 63, 90, 101, 136, 156, 201, 239, 296, 355, 428, 523, 617, 754, 878, 1078, 1243, 1517, 1741, 2121, 2426, 2928, 3348, 4021, 4596, 5468, 6257, 7400, 8472, 9936, 11389, 13285, 15233, 17645, 20244, 23346
OFFSET
0,5
LINKS
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
FORMULA
G.f.: (Product_{k>=1} (1+q^(2*k-1))/(1-q^(2*k))) - Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^(2*k))^2.
EXAMPLE
a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].
MAPLE
b:= proc(n, i, c) option remember; `if`(n=0,
`if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
`if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Jan 13 2021
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k - 1))/(1 - x^(2*k)), {k, 1, nmax/2}] - Sum[x^(k^2)/Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)
PROG
(PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(prod(k=1, N, (1+x^(2*k-1))/(1-x^(2*k)))-sum(k=0, sqrt(N), x^(k^2)/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jan 13 2021
STATUS
approved