OFFSET
7,1
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.
LINKS
Robert A. Russell, Table of n, a(n) for n = 7..100
Robert A. Russell, Trunk Generating Functions
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.
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Jun 16 2022
STATUS
approved