A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.

6, 21, 57, 138, 294, 606, 1170, 2208, 4008, 7176, 12492, 21510, 36348, 60801, 100281, 164019, 265263, 425853

Offset

3,1

Comments

Conjecture: equal to 3*(2*A000219 -A000990 -2*A000041 +1) tested up to n=20.

Links

Table of n, a(n) for n=3..20.

Wouter Meeussen, Mma functions for plane and solid partitions

Formula

G.f.: 3*(2*Product_{k>0} 1/(1-x^k)^k -(1-x)*Product_{k>0} 1/(1-x^k)^2 - 2*Product_{k>0} 1/(1-x^k) + 1/(1 - x)) (conjectured).

Example

a(3)=6 since the 6 solid partitions of {3,3} are:
  z[{{3}},{{3}}],
  z[{{2,1}},{{2,1}}],
  z[{{1,1,1}},{{1,1,1}}],z[{{2},{1}},{{2},{1}}],
  z[{{1,1},{1}},{{1,1},{1}}],
  z[{{1},{1},{1}},{{1},{1},{1}}].

Mathematica

Table[Length@solidformBTK[{n, 3}], {n, 3, 20}] (* or *)
g=20; 3 CoefficientList[Series[2/Product[(1-x^m)^m, {m, g}]+ 1/(1-x)-(1-x)/Product[(1-x^m)^2, {m, g}]-2/Product[(1-x^m), {m, g}], {x, 0, g}], x]

Crossrefs

Cf. A000041, A000219, A000990, A378854.

Sequence in context: A056341 A144899 A053809 * A290891 A047520 A392584

Adjacent sequences: A381262 A381263 A381264 * A381266 A381267 A381268

Keyword

nonn,more

Author

Wouter Meeussen, Feb 18 2025

Status

approved