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A340622 The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.

%I A340622 #16 May 29 2022 12:45:28
%S A340622 1,0,0,1,0,2,0,3,1,4,2,5,5,6,8,8,14,10,20,14,30,20,40,29,56,42,72,62,
%T A340622 96,88,122,125,160,174,202,239,263,322,334,431,434,566,554,739,719,
%U A340622 954,920,1222,1192,1552,1524,1964,1962,2466,2500,3088,3196,3848,4046
%N A340622 The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.
%H A340622 Vaclav Kotesovec, <a href="/A340622/b340622.txt">Table of n, a(n) for n = 0..6000</a>
%H A340622 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 A340622 G.f.: Sum_{n>=1} q^(n^2+2*n)/Product_{k=1..n} (1-q^(2*k))^2.
%F A340622 a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(1/2) * 5^(3/4) * (1 + sqrt(5)) * n). - _Vaclav Kotesovec_, Jan 14 2021
%e A340622 a(9) = 4 counts the partitions [8,1], [7,2], [6,3], and [5,4].
%p A340622 b:= proc(n, i, c) option remember; `if`(n=0,
%p A340622       `if`(c=0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
%p A340622       `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
%p A340622     end:
%p A340622 a:= n-> b(n$2, 0):
%p A340622 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 13 2021
%t A340622 b[n_, i_, c_] := b[n, i, c] = If[n == 0,
%t A340622      If[c == 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n-i*j, i-1, c + j*
%t A340622      If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
%t A340622 a[n_] := b[n, n, 0];
%t A340622 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 29 2022, after _Alois P. Heinz_ *)
%Y A340622 Cf. A006950, A340621, A340623.
%K A340622 nonn
%O A340622 0,6
%A A340622 _Jeremy Lovejoy_, Jan 13 2021