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

Author

Jeremy Lovejoy, Jan 15 2021

Keywords

• nonn

Examples

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

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.

Crossrefs

Cf. A001524, A015128, A340658.

Programs

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 *)

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

Extensions

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