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

Author

Jeremy Lovejoy, Jan 15 2021

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A001524, A015128, A340658, A340659.

Programs

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 *)

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))).