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

1, 1, 5, 17, 78, 375, 1919, 10144, 55189, 306632, 1734019, 9948977, 57790152, 339241199, 2009749140, 12002162624, 72186635028, 436913179401, 2659435211566, 16270345814930, 100002046716732, 617227859736748, 3824280874554199, 23778486784950053, 148329560863192846

Offset

1,3

Comments

Zero is not allowed as a part.

Signed partitions of n into n parts allowing zero as a part: A211475.

Links

Table of n, a(n) for n=1..25.

William J. Keith, A bijective toolkit for signed partitions, Ann. Combinat. 15 (1) (2011) 95-117.

Example

a(3) = 5: (1,1,1), (-1,1,3), (-1,2,2), (-2,2,3}, (-3,3,3).

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} minus {0}))))
    end:
a:= n-> b(n$4):
seq(a(n), n=1..15); # Alois P. Heinz, Apr 12 2012

Mathematica

Table[1 + Sum[Sum[(IntegerPartitions[k, {j}, Range[n]] // Length) * (IntegerPartitions[n + k, {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]~Complement~{0}}]]];
a[n_] := b[n, n, n, n];
Array[a, 25] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

Crossrefs

Cf. A211475.

Sequence in context: A330800 A293458 A009234 * A149744 A149745 A149746

Adjacent sequences: A211471 A211472 A211473 * A211475 A211476 A211477

Keyword

nonn

Author

David Scambler, Apr 12 2012

Extensions

More terms from Alois P. Heinz, Apr 12 2012

Status

approved