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

0,3

Comments

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A000041, A015128, A235792.

Sequence in context: A327285 A081996 A034329 * A340668 A133470 A129696

Adjacent sequences: A230438 A230439 A230440 * A230442 A230443 A230444

Keyword

nonn

Author

Omar E. Pol, Jan 09 2014

Status

approved