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

Author

Wouter Meeussen

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Equals (A000219+A005987)/2.

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

Programs

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 *)

Extensions

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