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

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, 96, 88, 122, 125, 160, 174, 202, 239, 263, 322, 334, 431, 434, 566, 554, 739, 719, 954, 920, 1222, 1192, 1552, 1524, 1964, 1962, 2466, 2500, 3088, 3196, 3848, 4046

Offset: 0

Author

Jeremy Lovejoy, Jan 13 2021

Keywords

• nonn

Examples

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
• B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Crossrefs

Cf. A006950, A340621, A340623.

Programs

Maple

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

Mathematica

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c == 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n-i*j, i-1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

Formula

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

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(1/2) * 5^(3/4) * (1 + sqrt(5)) * n). - Vaclav Kotesovec, Jan 14 2021