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

%I #12 Aug 01 2022 14:23:46

%S 8,76,440,2019,8147,30367,107061,361655,1181761,3762817,11733393,

%T 35957132,108591703,323914688,955984083,2795513143,8108894051,

%U 23354358683,66838785954,190211189706,538567451991,1517943035326

%N Number of chiral pairs of 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. Each member of a chiral pair is a reflection but not a rotation of the other.

%H Robert A. Russell, <a href="/A355049/b355049.txt">Table of n, a(n) for n = 7..100</a>

%H Robert A. Russell, <a href="/A355049/a355049.pdf">Trunk Generating Functions</a>

%F a(n) = A355047(n) - A355048(n) = (A355047(n) - A355050(n)) / 2 = A355048(n) - A355050(n).

%F G.f.: (14 C(x)^6 + 3 C(x)^7 + 6 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 18 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 10 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^3 (38 C(x)^4 + 9 C(x)^5 + 4 C(x)^2 C(-x^2) + 10 C(x)^3 C(-x^2) - 2 C(-x^2)^2 + C(x) C(-x^2)^2) / (8(1-C(x))) + C(x)^6 (5 C(x) + 16 C(x)^2 + 6 C(x)^3 + C(-x^2) + 2 C(x) C(-x^2)) / (2(1-C(x))^2) - C(-x^2)^2 (C(x)^4 + 2 C(x) C(-x^2) + 4 C(x)^2 C(-x^2) + 2 C(-x^2)^2 + 5 C(x) C(-x^2)^2 + C(-x^4) + C(x) C(-x^4)) / (4(1-C(-x^2))) + C(x)^7 (2 + 42 C(x) + 51 C(x)^2 + 24 C(x)^3 + 3 C(-x^2)) / (12(1-C(x))^3) + (C(x) C(x^3)^2) / (3(1-C(x^3))) - C(x)^2 C(-x^2)^2 (2 C(x) + 5 C(x)^3 + 2 C(-x^2) + C(x) C(-x^2)) / (4(1-C(x)) (1-C(-x^2))) - C(-x^2)^4 (8 + C(x) + 8 C(x) C(-x^2)) / (8(1-C(-x^2))^2) + C(x)^9 (17 + 8 C(x)) / (8(1-C(x))^4) - C(x)^5 (1 + 4 C(x)) C(-x^2)^2 / (4(1-C(x))^2 (1-C(-x^2))) + (C(x) C(-x^4)^2) / (4(1-C(-x^4))) + (3 C(x)^10) / (8(1-C(x))^5) - ((C(x)^6 C(-x^2)^2) / (4(1-C(x))^3 (1-C(-x^2)))) - (((1 + C(x)) C(-x^2)^5) / (4(1-C(-x^2))^3)) + ((1 + C(x)) C(-x^2) C(-x^4)^2) / (4(1-C(-x^2)) (1-C(-x^4))) - ((C(x)^2 C(-x^2)^4) / (8(1-C(x)) (1-C(-x^2))^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

%e a(7)=8 because there are 8 pairs of chiral heptominoes in 2^4 space. See trunks 1, 6, 8, 12, 13, 19, 27, and 28 in the linked Trunk Generating Functions.

%t sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];

%t c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);

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

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

%t Drop[CoefficientList[Series[(14 K[x]^6 + 3 K[x]^7 + 6 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 18 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 10 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^3 (38 K[x]^4 + 9 K[x]^5 + 4 K[x]^2 K[-x^2] + 10 K[x]^3 K[-x^2] - 2 K[-x^2]^2 + K[x] K[-x^2]^2) / (8(1-K[x])) + K[x]^6 (5 K[x] + 16 K[x]^2 + 6 K[x]^3 + K[-x^2] + 2 K[x] K[-x^2]) / (2(1-K[x])^2) - K[-x^2]^2 (K[x]^4 + 2 K[x] K[-x^2] + 4 K[x]^2 K[-x^2] + 2 K[-x^2]^2 + 5 K[x] K[-x^2]^2 + K[-x^4] + K[x] K[-x^4]) / (4(1-K[-x^2])) + K[x]^7 (2 + 42 K[x] + 51 K[x]^2 + 24 K[x]^3 + 3 K[-x^2]) / (12(1-K[x])^3) + (K[x] K[x^3]^2) / (3(1-K[x^3])) - K[x]^2 K[-x^2]^2 (2 K[x] + 5 K[x]^3 + 2 K[-x^2] + K[x] K[-x^2]) / (4(1-K[x]) (1-K[-x^2])) - K[-x^2]^4 (8 + K[x] + 8 K[x] K[-x^2]) / (8(1-K[-x^2])^2) + K[x]^9 (17 + 8 K[x]) / (8(1-K[x])^4) - K[x]^5 (1 + 4 K[x]) K[-x^2]^2 / (4(1-K[x])^2 (1-K[-x^2])) + (K[x] K[-x^4]^2) / (4(1-K[-x^4])) + (3 K[x]^10) / (8(1-K[x])^5) - ((K[x]^6 K[-x^2]^2) / (4(1-K[x])^3 (1-K[-x^2]))) - (((1 + K[x]) K[-x^2]^5) / (4(1-K[-x^2])^3)) + ((1 + K[x]) K[-x^2] K[-x^4]^2) / (4(1-K[-x^2]) (1-K[-x^4])) - ((K[x]^2 K[-x^2]^4) / (8(1-K[x]) (1-K[-x^2])^2)), {x,0,nmax}], x], 7]

%Y Cf. A355047 (oriented), A355048 (unoriented), A355050 (achiral) A355051 (asymmetric), A045648 (rooted chiral).

%Y Other dimensions: A036368 (n-2), A045649 (n-1), A355054 (multidimensional).

%K nonn,easy

%O 7,1

%A _Robert A. Russell_, Jun 16 2022