A378854 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 2.

3, 9, 24, 51, 111, 213, 414, 756, 1374, 2409, 4206, 7152, 12096, 20109, 33198, 54111, 87648, 140520, 223953, 354006, 556419, 868584, 1348857, 2082225, 3198927, 4888944, 7438548, 11265141, 16990077, 25516401, 38175240, 56894490, 84490935, 125028489, 184400952

Offset: 4

Author

Wouter Meeussen, Feb 10 2025

Keywords

• nonn

Comments

Appears to equal 3*A097391(n-3).

Examples

			a(4)=3 since the 3 solid partitions of {2,2} are:
  [{{2}},{{2}}], [{{1,1}},{{1,1}}], [{{1},{1}},{{1},{1}}]
a(5)=9 since the 9 solid partitions of {3,2} are:
  [{{3}},{{2}}], [{{2,1}},{{2}}], [{{2,1}},{{1,1}}], [{{1,1,1}},{{1,1}}], [{{2},{1}},{{2}}], [{{2},{1}},{{1},{1}}],[{{1,1},{1}},{{1,1}}], [{{1,1},{1}},{{1},{1}}], [{{1},{1},{1}},{{1},{1}}]
		

Links

• Wouter Meeussen Mma functions for plane and solid partitions

Crossrefs

Cf. A097391, A094504, A000219, A000041.

Programs

Mathematica

Table[Length@solidformBTK[{n-2,2}],{n,4,22}] (* uses functions defined in link above *)

PARI

A_x(N) = {my(x='x+O('x^N)); Vec(3*x^2*(prod(i=1,N, 1/(1-x^i)^i)-prod(i=1,N, 1/(1-x^i))))}
A_x(40) \\ John Tyler Rascoe, Feb 20 2025

Formula

From John Tyler Rascoe, Feb 20 2025: (Start)

a(n) = 3*(A000219(n-2) - A000041(n-2)).

G.f.: 3*x^2 * (Product_{i>0} (1/(1-x^i)^i) - Product_{i>0} (1/(1-x^i)^i)). (End)

Extensions

a(23) onwards from John Tyler Rascoe, Feb 20 2025