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The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.
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%I #11 May 14 2022 11:21:32

%S 0,0,1,2,5,9,17,29,49,79,125,193,293,437,642,932,1336,1896,2663,3709,

%T 5121,7020,9551,12913,17347,23172,30779,40679,53495,70030,91269,

%U 118459,153133,197214,253057,323595,412418,523953,663612,838035,1055304,1325287,1659969

%N 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 Also equal to A340658(n) - A001524(n).

%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^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 a(4) = 5 counts the overpartitions [3,1], [2,2], [2,1,1], [1,1,1,1], and [1',1,1,1].

%p b:= proc(n, i, c) option remember; `if`(n=0,

%p `if`(c>1, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(

%p add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 15 2021

%t b[n_, i_, c_] := b[n, i, c] = If[n == 0,

%t If[c > 1, 1, 0], If[i < 1, 0, b[n, i-1, c] + Sum[

%t Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];

%t a[n_] := b[n, n, 0];

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 14 2022, after _Alois P. Heinz_ *)

%Y Cf. A001524, A015128, A340658, A340659.

%K nonn

%O 0,4

%A _Jeremy Lovejoy_, Jan 15 2021