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

0,6

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.

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

Example

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

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 *)

Crossrefs

Cf. A006950, A340621, A340623.

Sequence in context: A084964 A267182 A008720 * A263352 A008734 A226649

Adjacent sequences: A340619 A340620 A340621 * A340623 A340624 A340625

Keyword

nonn

Author

Jeremy Lovejoy, Jan 13 2021

Status

approved