A230441 Number of overpartitions of n minus the number of partitions of n.

0, 1, 2, 5, 9, 17, 29, 49, 78, 124, 190, 288, 427, 627, 905, 1296, 1831, 2567, 3563, 4910, 6709, 9112, 12286, 16473, 21953, 29108, 38388, 50398, 65850, 85683, 111020, 143302, 184263, 236113, 301498, 383757, 486909, 615955, 776921, 977263, 1225934, 1533945

Offset: 0

Author

Omar E. Pol, Jan 09 2014

Keywords

• nonn

Comments

Number of overpartitions of n that contain at least one overlined part. - Omar E. Pol, Jan 19 2014

Examples

			The 14 overpartitions of 4 are
01: [4],
02: [4'],
03: [2, 2],
04: [2', 2],
05: [3, 1],
06: [3', 1],
07: [3, 1'],
08: [3', 1'],
09: [2, 1, 1],
10: [2', 1, 1],
11: [2, 1', 1],
12: [2', 1', 1],
13: [1, 1, 1, 1],
14: [1', 1, 1, 1].
There are 9 overpartitions that contain at least one overlined part, so a(4) = 9. - _Omar E. Pol_, Jan 19 2014
		

Links

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

Crossrefs

Cf. A000041, A015128, A235792.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0$2],
      b(n, i-1) +add((l->l+[0, l[2]])(b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> (l->l[2]-l[1])(b(n$2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {1, 1}, If[i<1, {0, 0}, b[n, i-1] + Sum[Function[ {l}, l+{0, l[[2]]}][b[n-i*j, i-1]], {j, 1, n/i}]]]; a[n_] := Function[{l}, l[[2]]-l[[1]]][b[n, n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 28 2015, after Alois P. Heinz *)

Formula

a(n) = A015128(n) - A000041(n).