%I #25 Jan 23 2021 18:30:22
%S 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,
%T 296,355,428,523,617,754,878,1078,1243,1517,1741,2121,2426,2928,3348,
%U 4021,4596,5468,6257,7400,8472,9936,11389,13285,15233,17645,20244,23346
%N The number of partitions of n without repeated odd parts having more even parts than odd parts.
%H Seiichi Manyama, <a href="/A340623/b340623.txt">Table of n, a(n) for n = 0..10000</a>
%H B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1016/j.ejc.2020.103159">Parity bias in partitions</a>, European J. Combin., 89 (2020), 103159, 19 pp.
%F 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.
%e a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].
%p b:= proc(n, i, c) option remember; `if`(n=0,
%p `if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
%p `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 13 2021
%t 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 *)
%o (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
%Y Cf. A006950, A340621, A340622, A340647.
%K nonn
%O 0,5
%A _Jeremy Lovejoy_, Jan 13 2021