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A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

%I A340659 #20 Jun 06 2021 05:48:20
%S A340659 1,0,1,2,3,5,7,11,15,23,31,45,61,85,114,156,206,276,363,477,621,808,
%T A340659 1041,1339,1713,2182,2769,3501,4409,5534,6927,8635,10741,13316,16467,
%U A340659 20303,24980,30643,37518,45815,55836,67889,82395,99772,120609,145501,175229,210637
%N A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.
%H A340659 Vaclav Kotesovec, <a href="/A340659/b340659.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H A340659 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 A340659 G.f.: Sum_{n>=0} q^(n*(n+1)/2 + n)/Product_{k=1..n} (1-q^k)^2.
%F A340659 a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * phi^2 * n), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Jun 06 2021
%F A340659 a(n) = A143184(n) - A001524(n). - _Vaclav Kotesovec_, Jun 06 2021
%e A340659 a(5) = 5 counts the overpartitions [4',1], [4,1'], [3',2], [3,2'], and [2',1',1,1].
%p A340659 b:= proc(n, i, c) option remember; `if`(n=0,
%p A340659       `if`(c=0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
%p A340659        add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
%p A340659     end:
%p A340659 a:= n-> b(n$2, 0):
%p A340659 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 15 2021
%t A340659 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 A340659 a[n_] := b[n, n, 0];
%t A340659 a /@ Range[0, 60] (* _Jean-François Alcover_, Jan 29 2021, after _Alois P. Heinz_ *)
%t A340659 nmax = 50; CoefficientList[Series[1 + Sum[x^(j*(j+1)/2 + j) / QPochhammer[x, x, j]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 06 2021 *)
%Y A340659 Cf. A001524, A015128, A340658.
%K A340659 nonn
%O A340659 0,4
%A A340659 _Jeremy Lovejoy_, Jan 15 2021
%E A340659 a(0)=1 prepended by _Alois P. Heinz_, Jan 15 2021