A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

0, 0, 1, 2, 5, 9, 17, 29, 49, 79, 125, 193, 293, 437, 642, 932, 1336, 1896, 2663, 3709, 5121, 7020, 9551, 12913, 17347, 23172, 30779, 40679, 53495, 70030, 91269, 118459, 153133, 197214, 253057, 323595, 412418, 523953, 663612, 838035, 1055304, 1325287, 1659969

Offset

0,4

Comments

Also equal to A340658(n) - A001524(n).

Links

Table of n, a(n) for n=0..42.

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

Formula

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

Example

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

Maple

b:= proc(n, i, c) option remember; `if`(n=0,
     `if`(c>1, 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 > 1, 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];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

Crossrefs

Cf. A001524, A015128, A340658, A340659.

Sequence in context: A081996 A034329 A230441 * A133470 A129696 A082281

Adjacent sequences: A340665 A340666 A340667 * A340669 A340670 A340671

Keyword

nonn

Author

Jeremy Lovejoy, Jan 15 2021

Status

approved