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

1, 0, 1, 2, 3, 5, 7, 11, 15, 23, 31, 45, 61, 85, 114, 156, 206, 276, 363, 477, 621, 808, 1041, 1339, 1713, 2182, 2769, 3501, 4409, 5534, 6927, 8635, 10741, 13316, 16467, 20303, 24980, 30643, 37518, 45815, 55836, 67889, 82395, 99772, 120609, 145501, 175229, 210637

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Formula

G.f.: Sum_{n>=0} q^(n*(n+1)/2 + n)/Product_{k=1..n} (1-q^k)^2.

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

a(n) = A143184(n) - A001524(n). - Vaclav Kotesovec, Jun 06 2021

Example

a(5) = 5 counts the overpartitions [4',1], [4,1'], [3',2], [3,2'], and [2',1',1,1].

Maple

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
       add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

Mathematica

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}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)
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 *)

Crossrefs

Cf. A001524, A015128, A340658.

Sequence in context: A366845 A024792 A280661 * A055771 A052955 A326466

Adjacent sequences: A340656 A340657 A340658 * A340660 A340661 A340662

Keyword

nonn

Author

Jeremy Lovejoy, Jan 15 2021

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 15 2021

Status

approved