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A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

%I A340668 #11 May 14 2022 11:21:32
%S A340668 0,0,1,2,5,9,17,29,49,79,125,193,293,437,642,932,1336,1896,2663,3709,
%T A340668 5121,7020,9551,12913,17347,23172,30779,40679,53495,70030,91269,
%U A340668 118459,153133,197214,253057,323595,412418,523953,663612,838035,1055304,1325287,1659969
%N A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.
%C A340668 Also equal to A340658(n) - A001524(n).
%H A340668 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 A340668 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) * (1-q^(n+1))).
%e A340668 a(4) = 5 counts the overpartitions [3,1], [2,2], [2,1,1], [1,1,1,1], and [1',1,1,1].
%p A340668 b:= proc(n, i, c) option remember; `if`(n=0,
%p A340668      `if`(c>1, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
%p A340668       add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
%p A340668     end:
%p A340668 a:= n-> b(n$2, 0):
%p A340668 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 15 2021
%t A340668 b[n_, i_, c_] := b[n, i, c] = If[n == 0,
%t A340668      If[c > 1, 1, 0], If[i < 1, 0, b[n, i-1, c] + Sum[
%t A340668      Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
%t A340668 a[n_] := b[n, n, 0];
%t A340668 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 14 2022, after _Alois P. Heinz_ *)
%Y A340668 Cf. A001524, A015128, A340658, A340659.
%K A340668 nonn
%O A340668 0,4
%A A340668 _Jeremy Lovejoy_, Jan 15 2021