A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

0, 1, 2, 4, 8, 14, 25, 41, 67, 105, 163, 246, 368, 540, 784, 1124, 1596, 2242, 3124, 4316, 5918, 8058, 10899, 14651, 19581, 26028, 34417, 45293, 59327, 77372, 100483, 129984, 167502, 215077, 275199, 350966, 446162, 565451, 714515, 900334, 1131370, 1417963

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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.

Example

a(3) = 4 counts the overpartitions [3], [2,1], [1,1,1], and [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 *)

Crossrefs

Cf. A001524, A015128, A340659.

Sequence in context: A164175 A066456 A066342 * A291443 A210145 A020956

Adjacent sequences: A340655 A340656 A340657 * A340659 A340660 A340661

Keyword

nonn

Author

Jeremy Lovejoy, Jan 15 2021

Status

approved