A211475 Signed partitions of n into n parts in {-n..n} (allowing zero as a part).

1, 2, 7, 27, 121, 587, 2983, 15744, 85375, 473259, 2670383, 15293119, 88686530, 519864702, 3075894246, 18348407371, 110244289384, 666651567920, 4054481396896, 24786629709850, 152241407914480, 939069024577371, 5815027475345028, 36137289604644570

Offset

1,2

Comments

The normal definition of signed partitions does not allow zero as a part.

Signed partitions of n into n parts in {-n..n}\{0}: A211474.

Links

Table of n, a(n) for n=1..24.

Example

a(3) = 7: (-3,3,3), (-2,2,3), (-1,1,3), (-1,2,2), (0,0,3), (0,1,2), (1,1,1).

Maple

b:= proc(h, i, t, n) option remember;
      `if`(i=0, `if`(h=0, 1, 0), `if`(h<0 or i*n<h, 0,
       add (b(h+j, i-1, j, n), j=-n..t)))
    end:
a:= n-> b(n$4):
seq (a(n), n=1..15); # Alois P. Heinz, Apr 12 2012

Mathematica

Table[(IntegerPartitions[n, {1, n}] // Length) + Sum[Sum[(IntegerPartitions[k, {j}, Range[n]] // Length) * (IntegerPartitions[n + k, {1, n - j}, Range[n]] // Length), {j, 0, n - 2}], {k, 1, n*Floor[(n - 1)/2]}], {n, 14}]
(* Second program: *)
b[h_, i_, t_, n_] := b[h, i, t, n] =
     If[i == 0, If[h == 0, 1, 0], If[h < 0 || i*n < h, 0,
     Sum[b[h + j, i - 1, j, n], {j, Range[-n, t]}]]];
a[n_] := b[n, n, n, n];
Array[a, 24] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

Crossrefs

Cf. A211474.

Sequence in context: A368762 A150645 A060017 * A213226 A058800 A357901

Adjacent sequences: A211472 A211473 A211474 * A211476 A211477 A211478

Keyword

nonn

Author

David Scambler, Apr 12 2012

Extensions

More terms from Alois P. Heinz, Apr 12 2012

Status

approved