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Number of asymmetric orthoplex n-ominoes with cell centers determining n-3 space.
7

%I #9 Aug 01 2022 14:23:41

%S 6,67,412,1926,7856,29057,101105,335081,1072653,3337131,10154700,

%T 30330869,89226443,259092076,744095757,2116643127,5971171140,

%U 16722250081,46529076097,128722040503,354276958783,970546150818

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

%C 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.

%H Robert A. Russell, <a href="/A355051/b355051.txt">Table of n, a(n) for n = 7..100</a>

%H Robert A. Russell, <a href="/A355051/a355051.pdf">Trunk Generating Functions</a>

%F 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.

%e 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.

%t sa[n_, k_] := sa[n, k] = a[n+1-k, 1] + If[n < 2 k, 0, -sa[n-k, k]];

%t 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);

%t a[n_, k_] := a[n, k] = Sum[a[i, 1] a[n-i, k-1], {i, 1, n-1}];

%t nmax = 30; A[x_] := Sum[a[i, 1] x^i, {i, 0, nmax}]

%t 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]

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

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

%K nonn,easy

%O 7,1

%A _Robert A. Russell_, Jun 16 2022