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A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

%I A340658 #14 Jan 29 2021 14:49:05
%S A340658 0,1,2,4,8,14,25,41,67,105,163,246,368,540,784,1124,1596,2242,3124,
%T A340658 4316,5918,8058,10899,14651,19581,26028,34417,45293,59327,77372,
%U A340658 100483,129984,167502,215077,275199,350966,446162,565451,714515,900334,1131370,1417963
%N A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.
%H A340658 Alois P. Heinz, <a href="/A340658/b340658.txt">Table of n, a(n) for n = 0..1000</a>
%H A340658 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 A340658 G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/Product_{k=1..n} (1-q^k)^2.
%e A340658 a(3) = 4 counts the overpartitions [3], [2,1], [1,1,1], and [1',1,1].
%p A340658 b:= proc(n, i, c) option remember; `if`(n=0,
%p A340658      `if`(c>0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
%p A340658       add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
%p A340658     end:
%p A340658 a:= n-> b(n$2, 0):
%p A340658 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 15 2021
%t A340658 b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c>0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
%t A340658 a[n_] := b[n, n, 0];
%t A340658 a /@ Range[0, 60] (* _Jean-François Alcover_, Jan 29 2021, after _Alois P. Heinz_ *)
%Y A340658 Cf. A001524, A015128, A340659.
%K A340658 nonn
%O A340658 0,3
%A A340658 _Jeremy Lovejoy_, Jan 15 2021