A355051 Number of asymmetric orthoplex n-ominoes with cell centers determining n-3 space.

6, 67, 412, 1926, 7856, 29057, 101105, 335081, 1072653, 3337131, 10154700, 30330869, 89226443, 259092076, 744095757, 2116643127, 5971171140, 16722250081, 46529076097, 128722040503, 354276958783, 970546150818

Offset: 7

Author

Robert A. Russell, Jun 16 2022

Keywords

• nonn
• easy

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. An asymmetric polyomino has a symmetry group of order 1.

Examples

			a(7)=6 because there are 6 asymmetric heptominoes in 2^4 space. See trunks 1, 6, 8, 12, 27, and 28 in the linked Trunk Generating Functions.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 7..100
• Robert A. Russell, Trunk Generating Functions

Crossrefs

Cf. A355047 (oriented), A355048 (unoriented), A355049 (chiral) A355050 (achiral), A004111 (rooted asymmetric).

Other dimensions: A036369 (n-2), A000220 (n-1), A355056 (multidimensional).

Programs

Mathematica

sa[n_, k_] := sa[n, k] = a[n+1-k, 1] + If[n < 2 k, 0, -sa[n-k, k]];
a[1, 1] := 1; a[n_, 1] := a[n, 1] = Sum[a[i, 1] sa[n-1, i] i, {i, 1, n-1}]/(n-1);
a[n_, k_] := a[n, k] = Sum[a[i, 1] a[n-i, k-1], {i, 1, n-1}];
nmax = 30; A[x_] := Sum[a[i, 1] x^i, {i, 0, nmax}]
Drop[CoefficientList[Series[(14 A[x]^6 + 103 A[x]^7 + 24 A[x]^8 - 6 A[x]^4 A[x^2] - 12 A[x]^5 A[x^2] - 24 A[x]^6 A[x^2] - 18 A[x]^2 A[x^2]^2 + 15 A[x]^3 A[x^2]^2 - 14 A[x^2]^3 + 8 A[x] A[x^2]^3 + 6 A[x]^2 A[x^2]^3 + 4 A[x^3]^2 - 4 A[x] A[x^3]^2 + 24 A[x^2] A[x^4] - 18 A[x] A[x^2] A[x^4] - 6 A[x]^2 A[x^2] A[x^4] - 4 A[x^6] + 4 A[x] A[x^6])/(24 (1-A[x])) + A[x]^6 (5 A[x] + 16 A[x]^2 + 6 A[x]^3 - A[x^2] - 2 A[x] A[x^2])/(2 (1-A[x])^2) - A[x^2] (A[x]^4 A[x^2] + 8 A[x] A[x^2]^2 + 2 A[x]^2 A[x^2]^2 + 10 A[x^2]^3 + 5 A[x] A[x^2]^3 - 2 A[x] A[x^4] - 3 A[x^2] A[x^4] - A[x] A[x^2] A[x^4])/(4 (1-A[x^2])) + A[x]^7 (2 + 42 A[x] + 51 A[x]^2 + 24 A[x]^3 - 3 A[x^2])/(12 (1-A[x])^3) - A[x]^2 A[x^2]^2 (2 A[x] + 5 A[x]^3 + 2 A[x^2] - A[x] A[x^2])/(4 (1-A[x]) (1-A[x^2])) + A[x] A[x^3]^2/(1-A[x^3])/3 + A[x]^9 (17 + 8 A[x])/(8 (1-A[x])^4) - A[x]^5 (1 + 4 A[x]) A[x^2]^2/(4 (1-A[x])^2 (1-A[x^2])) - A[x^2]^4 (8 + 17 A[x] + 16 A[x^2] + 8 A[x] A[x^2])/(8 (1-A[x^2])^2) + A[x] (A[x^4]^2/(1-A[x^4]))/4 + 3 A[x]^10/(8 (1-A[x])^5) - A[x]^6 A[x^2]^2/(4 (1-A[x])^3 (1-A[x^2])) - A[x]^2 A[x^2]^4/(8 (1-A[x]) (1-A[x^2])^2) - 3 (1 + A[x]) A[x^2]^5/(4 (1-A[x^2])^3) +3 (1 + A[x]) A[x^2] A[x^4]^2/(4 (1-A[x^2]) (1-A[x^4])), {x,0,nmax}], x], 7]

Formula

G.f.: (14 A(x)^6 + 103 A(x)^7 + 24 A(x)^8 - 6 A(x)^4 A(x^2) - 12 A(x)^5 A(x^2) - 24 A(x)^6 A(x^2) - 18 A(x)^2 A(x^2)^2 + 15 A(x)^3 A(x^2)^2 - 14 A(x^2)^3 + 8 A(x) A(x^2)^3 + 6 A(x)^2 A(x^2)^3 + 4 A(x^3)^2 - 4 A(x) A(x^3)^2 + 24 A(x^2) A(x^4) - 18 A(x) A(x^2) A(x^4) - 6 A(x)^2 A(x^2) A(x^4) - 4 A(x^6) + 4 A(x) A(x^6))/(24 (1 - A(x))) +A(x)^6 (5 A(x) + 16 A(x)^2 + 6 A(x)^3 - A(x^2) - 2 A(x) A(x^2))/(2 (1 - A(x))^2) - A(x^2) (A(x)^4 A(x^2) + 8 A(x) A(x^2)^2 + 2 A(x)^2 A(x^2)^2 + 10 A(x^2)^3 + 5 A(x) A(x^2)^3 - 2 A(x) A(x^4) - 3 A(x^2) A(x^4) - A(x) A(x^2) A(x^4))/(4 (1 - A(x^2))) + A(x)^7 (2 + 42 A(x) + 51 A(x)^2 + 24 A(x)^3 - 3 A(x^2))/(12 (1 - A(x))^3) - A(x)^2 A(x^2)^2 (2 A(x) + 5 A(x)^3 + 2 A(x^2) - A(x) A(x^2))/(4 (1 - A(x)) (1 - A(x^2))) + A(x) A(x^3)^2/(1 - A(x^3))/3 + A(x)^9 (17 + 8 A(x))/(8 (1 - A(x))^4) - A(x)^5 (1 + 4 A(x)) A(x^2)^2/(4 (1 - A(x))^2 (1 - A(x^2))) - A(x^2)^4 (8 + 17 A(x) + 16 A(x^2) + 8 A(x) A(x^2))/(8 (1 - A(x^2))^2) + A(x) (A(x^4)^2/(1 - A(x^4)))/4 + 3 A(x)^10/(8 (1 - A(x))^5) - A(x)^6 A(x^2)^2/(4 (1 - A(x))^3 (1 - A(x^2))) - A(x)^2 A(x^2)^4/(8 (1 - A(x)) (1 - A(x^2))^2) - 3 (1 + A(x)) A(x^2)^5/(4 (1 - A(x^2))^3) + 3 (1 + A(x)) A(x^2) A(x^4)^2/(4 (1 - A(x^2)) (1 - A(x^4))) where A(x) is the generating function for rooted identity trees with n nodes in A004111.