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The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.
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%I #16 May 29 2022 12:45:28

%S 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 96,88,122,125,160,174,202,239,263,322,334,431,434,566,554,739,719,

%U 954,920,1222,1192,1552,1524,1964,1962,2466,2500,3088,3196,3848,4046

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

%H Vaclav Kotesovec, <a href="/A340622/b340622.txt">Table of n, a(n) for n = 0..6000</a>

%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.: Sum_{n>=1} q^(n^2+2*n)/Product_{k=1..n} (1-q^(2*k))^2.

%F a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(1/2) * 5^(3/4) * (1 + sqrt(5)) * n). - _Vaclav Kotesovec_, Jan 14 2021

%e a(9) = 4 counts the partitions [8,1], [7,2], [6,3], and [5,4].

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

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

%p `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))

%p end:

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

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jan 13 2021

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

%t If[c == 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n-i*j, i-1, c + j*

%t If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];

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

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 29 2022, after _Alois P. Heinz_ *)

%Y Cf. A006950, A340621, A340623.

%K nonn

%O 0,6

%A _Jeremy Lovejoy_, Jan 13 2021