A048140 Number of planar partitions of n, but partitions that are mirror images of each other (when regarded as 3-D objects) are counted only once.

1, 2, 4, 8, 14, 27, 47, 86, 149, 261, 444, 760, 1269, 2119, 3486, 5711, 9247, 14906, 23800, 37816, 59622, 93528, 145759, 226071, 348612, 535131, 817280, 1242824, 1881310, 2836377, 4258509, 6369669, 9491142, 14092537, 20851146, 30749471

Offset

1,2

Comments

Plane partitions seen as 3-dimensional-objects can have a mirror symmetry plane.

Links

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Example

n=3 gives 4 forms: {{3}}; {{1,1,1}}={{1},{1},{1}}; {{2,1}}={{2},{1}}; {{1,1},{1}}.

Mathematica

terms = 100;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[ (terms+1)/2]}] + O[x]^(terms+1);
A005987 = CoefficientList[s, x];
a[n_] := (a219[n] + A005987[[n+1]])/2;
a /@ Range[terms] (* Jean-François Alcover, Dec 28 2019 *)

Crossrefs

Equals (A000219+A005987)/2.

Equals 2 Cs + 3 C1 + C3 + C3v, Cs=A000784, C1=A000785, C3=A048142, C3v=A048141. Cf. A000219, A005987.

Sequence in context: A164161 A068011 A048238 * A179817 A214255 A065616

Adjacent sequences: A048137 A048138 A048139 * A048141 A048142 A048143

Keyword

nonn

Author

Wouter Meeussen

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

Status

approved