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%I #12 Aug 01 2022 14:23:56
%S 3,18,122,655,3240,14531,61520,247381,958434,3598594,13180348,
%T 47274577,166642096,578750970,1984671466,6731351834,22612409886,
%U 75321920403,249028297179,817867225710,2670093233760,8670380548402
%N Number of unoriented 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. For unoriented polyominoes, chiral pairs are counted as one.
%H Robert A. Russell, <a href="/A355048/b355048.txt">Table of n, a(n) for n = 6..100</a>
%H Robert A. Russell, <a href="/A355048/a355048_1.pdf">Trunk Generating Functions</a>
%F a(n) = A355047(n) - A355049(n) = (A355047(n) + A355050(n)) / 2 = A355049(n) + A355050(n).
%F G.f.: (14B(x}^6 + 3B(x}^7 + 6B(x}^4B(x^2} + 6B(x}^5B(x^2} + 18B(x}^2B(x^2}^2 + 3B(x}^3B(x^2}^2 + 26B(x^2}^3 + 6 B(x}B(x^2}(B(x^2}^2 + B(x^4}) + 4B(x^3}^2 + 4B(x^6}) / 24 + B(x}^3 (38B(x}^4 + 9B(x}^5 + 4B(x}^2B(x^2} + 10B(x}^3B(x^2} + 2B(x^2}^2 + B(x}B(x^2}^2) / (8(1-B(x})) + B(x}^6 (16B(x}^2 + 6B(x}^3 + B(x^2} + B(x} (5 + 2B(x^2})) / (2(1-B(x})^2) + B(x}^7 (2 + 42B(x} + 51B(x}^2 + 24B(x}^3 + 3B(x^2}) / (12(1-B(x})^3) + B(x}^9 (17 + 8B(x}) / (8(1-B(x})^4) + 3B(x}^10 / (8(1-B(x})^5) + B(x^2}^2(B(x}^4 + 4B(x}^2 B(x^2} + 12B(x^2}^2 + B(x^4} + B(x} (8B(x^2} + 5B(x^2}^2 + B(x^4})) / (4(1-B(x^2})) + B(x^2}^4 (8 + 16B(x^2} + B(x} (19 + 8B(x^2})) / (8(1-B(x^2})^2) + 3(1 + B(x})B(x^2}^5 / (4(1-B(x^2})^3) + 2B(x}B(x^3}^2 / (6(1-B(x^3})) + B(x}B(x^4}^2 / (4(1-B(x^4})) + B(x}^2B(x^2}^2(5B(x}^3 + 2B(x^2} + B(x}(2 + B(x^2})) / (4(1-B(x})(1-B(x^2})) + B(x}^5(1+4B(x})B(x^2}^2 / (4(1-B(x})^2(1-B(x^2})) + B(x}^6 B(x^2}^2 / (4(1-B(x})^3(1-B(x^2})) + 3B(x}^2B(x^2}^4 / (8(1-B(x})(1-B(x^2})^2) + B(x^2}(1+B(x})B(x^4}^2 / (4(1-B(x^2})(1-B(x^4})), where B(x) is the generating function for rooted trees with n nodes in A000081.
%e a(6)=3 because there are 3 hexominoes in 2^3 space. The two vacant cells share just a face, an edge, or a vertex.
%t sb[n_,k_] := sb[n,k] = b[n+1-k,1] + If[n<2k,0,sb[n-k,k]];
%t b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
%t b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
%t nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
%t Drop[CoefficientList[Series[(14B[x]^6 + 3B[x]^7 + 6B[x]^4B[x^2] + 6B[x]^5B[x^2] + 18B[x]^2B[x^2]^2 + 3B[x]^3B[x^2]^2 + 26B[x^2]^3 + 6 B[x]B[x^2](B[x^2]^2 + B[x^4]) + 4B[x^3]^2 + 4B[x^6]) / 24 + B[x]^3 (38B[x]^4 + 9B[x]^5 + 4B[x]^2B[x^2] + 10B[x]^3B[x^2] + 2B[x^2]^2 + B[x]B[x^2]^2) / (8(1-B[x])) + B[x]^6 (16B[x]^2 + 6B[x]^3 + B[x^2] + B[x] (5 + 2B[x^2])) / (2(1-B[x])^2) + B[x]^7 (2 + 42B[x] + 51B[x]^2 + 24B[x]^3 + 3B[x^2]) / (12(1-B[x])^3) + B[x]^9 (17 + 8B[x]) / (8(1-B[x])^4) + 3B[x]^10 / (8(1-B[x])^5) + B[x^2]^2(B[x]^4 + 4B[x]^2 B[x^2] + 12B[x^2]^2 + B[x^4] + B[x] (8B[x^2] + 5B[x^2]^2 + B[x^4])) / (4(1-B[x^2])) + B[x^2]^4 (8 + 16B[x^2] + B[x] (19 + 8B[x^2])) / (8(1-B[x^2])^2) + 3(1 + B[x])B[x^2]^5 / (4(1-B[x^2])^3) + 2B[x]B[x^3]^2 / (6(1-B[x^3])) + B[x]B[x^4]^2 / (4(1-B[x^4])) + B[x]^2B[x^2]^2(5B[x]^3 + 2B[x^2] + B[x](2 + B[x^2])) / (4(1-B[x])(1-B[x^2])) + B[x]^5(1+4B[x])B[x^2]^2 / (4(1-B[x])^2(1-B[x^2])) + B[x]^6 B[x^2]^2 / (4(1-B[x])^3(1-B[x^2])) + 3B[x]^2B[x^2]^4 / (8(1-B[x])(1-B[x^2])^2) + B[x^2](1+B[x])B[x^4]^2 / (4(1-B[x^2])(1-B[x^4])), {x,0,nmax}],x],6]
%Y Cf. A355047 (oriented), A355049 (chiral), A355050 (achiral) A355051 (asymmetric), A000081 (rooted trees).
%Y Other dimensions: A036367 (n-2), A000055 (n-1), A355053 (multidimensional).
%K nonn,easy
%O 6,1
%A _Robert A. Russell_, Jun 16 2022
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