cp's OEIS Frontend

A355048 Number of unoriented orthoplex n-ominoes with cell centers determining n-3 space.

%I A355048 #15 Mar 22 2025 20:43:22
%S A355048 3,18,122,655,3240,14531,61520,247381,958434,3598594,13180348,
%T A355048 47274577,166642096,578750970,1984671466,6731351834,22612409886,
%U A355048 75321920403,249028297179,817867225710,2670093233760,8670380548402
%N A355048 Number of unoriented orthoplex n-ominoes with cell centers determining n-3 space.
%C A355048 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 A355048 Robert A. Russell, <a href="/A355048/b355048.txt">Table of n, a(n) for n = 6..100</a>
%H A355048 Robert A. Russell, <a href="/A355048/a355048_1.pdf">Trunk Generating Functions</a>
%F A355048 a(n) = A355047(n) - A355049(n) = (A355047(n) + A355050(n)) / 2 = A355049(n) + A355050(n).
%F A355048 G.f.: (14*B(x)^6 + 3*B(x)^7 + 6*B(x)^4*B(x^2) + 6*B(x)^5*B(x^2) + 18*B(x)^2*B(x^2)^2 + 3*B(x)^3*B(x^2)^2 + 26*B(x^2)^3 + 6*B(x)*B(x^2)*(B(x^2)^2 + B(x^4)) + 4*B(x^3)^2 + 4*B(x^6)) / 24 + B(x)^3*(38*B(x)^4 + 9*B(x)^5 + 4*B(x)^2*B(x^2) + 10*B(x)^3*B(x^2) + 2*B(x^2)^2 + B(x)*B(x^2)^2) / (8*(1-B(x))) + B(x)^6*(16*B(x)^2 + 6*B(x)^3 + B(x^2) + B(x)*(5 + 2*B(x^2))) / (2*(1-B(x))^2) + B(x)^7*(2 + 42*B(x) + 51*B(x)^2 + 24*B(x)^3 + 3*B(x^2)) / (12*(1-B(x))^3) + B(x)^9*(17 + 8*B(x)) / (8*(1-B(x))^4) + 3*B(x)^10 / (8*(1-B(x))^5) + B(x^2)^2*(B(x)^4 + 4*B(x)^2*B(x^2) + 12*B(x^2)^2 + B(x^4) + B(x)*(8*B(x^2) + 5*B(x^2)^2 + B(x^4))) / (4*(1-B(x^2))) + B(x^2)^4*(8 + 16*B(x^2) + B(x)*(19 + 8*B(x^2))) / (8*(1-B(x^2))^2) + 3*(1 + B(x))*B(x^2)^5 / (4*(1-B(x^2))^3) + 2*B(x)*B(x^3)^2 / (6*(1-B(x^3))) + B(x)*B(x^4)^2 / (4*(1-B(x^4))) + B(x)^2*B(x^2)^2*(5*B(x)^3 + 2*B(x^2) + B(x)*(2 + B(x^2))) / (4*(1-B(x))*(1-B(x^2))) + B(x)^5*(1+4*B(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))) + 3*B(x)^2*B(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 A355048 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 A355048 sb[n_,k_] := sb[n,k] = b[n+1-k,1] + If[n<2k,0,sb[n-k,k]];
%t A355048 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 A355048 b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
%t A355048 nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
%t A355048 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 A355048 Cf. A355047 (oriented), A355049 (chiral), A355050 (achiral) A355051 (asymmetric), A000081 (rooted trees).
%Y A355048 Other dimensions: A036367 (n-2), A000055 (n-1), A355053 (multidimensional).
%K A355048 nonn,easy
%O A355048 6,1
%A A355048 _Robert A. Russell_, Jun 16 2022