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

Author

Jeremy Lovejoy, Jan 15 2021

Keywords

• nonn

Examples

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

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.

Crossrefs

Cf. A001524, A015128, A340659.

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 *)

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.