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A355056
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Number of asymmetric multidimensional n-ominoes with cell centers determining n-3 space.
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6
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5, 46, 275, 1283, 5281, 19607, 68476, 227196, 727780, 2263148, 6881482, 20529511, 60312548, 174870492, 501443277, 1424142358, 4011274417, 11216074419, 31160837273, 86078096135, 236568911194, 647181951619
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OFFSET
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5,1
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COMMENTS
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Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). An asymmetric polyomino has a symmetry group of order 1.
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LINKS
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FORMULA
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G.f.: (4 A(x)^4 + 37 A(x)^5 + 12 A(x)^6 - 6 A(x)^3 A(x^2) - 10 A(x)^4 A(x^2) - 4 A(x^2)^2 - 17 A(x) A(x^2)^2 - 2 A(x^2)^3 + 2 A(x) A(x^4)) / 8 + (24 A(x)^5 + 515 A(x)^6 + 325 A(x)^7 + 24 A(x)^8 - 48 A(x)^4 A(x^2) - 96 A(x)^5 A(x^2) - 24 A(x)^6 A(x^2) - 21 A(x)^2 A(x^2)^2 + 21 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)^5 (2 A(x) + 67 A(x)^2 + 46 A(x)^3 + 6 A(x)^4 - 3 A(x^2) - 6 A(x) A(x^2) - 2 A(x)^2 A(x^2)) / (2 (1-A(x))^2) - A(x^2) (2 A(x)^2 A(x^2) + 6 A(x)^3 A(x^2) + 2 A(x)^4 A(x^2) + 13 A(x^2)^2 + 31 A(x) A(x^2)^2 + 2 A(x)^2 A(x^2)^2 + 15 A(x^2)^3 + 5 A(x) A(x^2)^3 - 3 A(x^4) - 5 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)^6 (4 A(x) + 153 A(x)^2 + 75 A(x)^3 + 12 A(x)^4 - 3 A(x^2) - 3 A(x) A(x^2)) / (6 (1-A(x))^3) - A(x)^2 A(x^2)^2 (2 A(x) + 7 A(x)^2 + 5 A(x)^3 + A(x^2) - A(x) A(x^2)) / (2 (1-A(x)) (1-A(x^2))) + A(x) A(x^3)^2 / (1-A(x^3)) / 3 + A(x)^9 (21 + 4 A(x)) / (2 (1-A(x))^4) - A(x)^5 (3 + 2 A(x)) A(x^2)^2 / ((1-A(x))^2 (1-A(x^2))) - A(x^2)^4 (5 + 7 A(x) + 3 A(x^2) + A(x) A(x^2)) / (1-A(x^2))^2 + A(x) A(x^4)^2 / (2 (1-A(x^4))) + 3 A(x)^10 / (2 (1-A(x))^5) - A(x)^6 A(x^2)^2 / ((1-A(x))^3 (1-A(x^2))) - 2 (1 + A(x)) A(x^2)^5 / (1-A(x^2))^3 + 3 (1 + A(x)) A(x^2) A(x^4)^2 / (2 (1-A(x^2)) (1-A(x^4))) where A(x) is the generating function for rooted identity trees with n nodes in A004111.
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EXAMPLE
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a(5)=5 as there are exactly five asymmetric pentominoes in 2-space.
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MATHEMATICA
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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[(4 A[x]^4 + 37 A[x]^5 + 12 A[x]^6 - 6 A[x]^3 A[x^2] - 10 A[x]^4 A[x^2] - 4 A[x^2]^2 - 17 A[x] A[x^2]^2 - 2 A[x^2]^3 + 2 A[x] A[x^4]) / 8 + (24 A[x]^5 + 515 A[x]^6 + 325 A[x]^7 + 24 A[x]^8 - 48 A[x]^4 A[x^2] - 96 A[x]^5 A[x^2] - 24 A[x]^6 A[x^2] - 21 A[x]^2 A[x^2]^2 + 21 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]^5 (2 A[x] + 67 A[x]^2 + 46 A[x]^3 + 6 A[x]^4 - 3 A[x^2] - 6 A[x] A[x^2] - 2 A[x]^2 A[x^2]) / (2 (1-A[x])^2) - A[x^2] (2 A[x]^2 A[x^2] + 6 A[x]^3 A[x^2] + 2 A[x]^4 A[x^2] + 13 A[x^2]^2 + 31 A[x] A[x^2]^2 + 2 A[x]^2 A[x^2]^2 + 15 A[x^2]^3 + 5 A[x] A[x^2]^3 - 3 A[x^4] - 5 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]^6 (4 A[x] + 153 A[x]^2 + 75 A[x]^3 + 12 A[x]^4 - 3 A[x^2] - 3 A[x] A[x^2]) / (6 (1-A[x])^3) - A[x]^2 A[x^2]^2 (2 A[x] + 7 A[x]^2 + 5 A[x]^3 + A[x^2] - A[x] A[x^2]) / (2 (1-A[x]) (1-A[x^2])) + A[x] A[x^3]^2 / (1-A[x^3]) / 3 + A[x]^9 (21 + 4 A[x]) / (2 (1-A[x])^4) - A[x]^5 (3 + 2 A[x]) A[x^2]^2 / ((1-A[x])^2 (1-A[x^2])) - A[x^2]^4 (5 + 7 A[x] + 3 A[x^2] + A[x] A[x^2]) / (1-A[x^2])^2 + A[x] A[x^4]^2 / (2 (1-A[x^4])) + 3 A[x]^10 / (2 (1-A[x])^5) - A[x]^6 A[x^2]^2 / ((1-A[x])^3 (1-A[x^2])) - 2 (1 + A[x]) A[x^2]^5 / (1-A[x^2])^3 + 3 (1 + A[x]) A[x^2] A[x^4]^2 / (2 (1-A[x^2]) (1-A[x^4])), {x, 0, nmax}], x], 5]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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