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%I #8 Aug 16 2016 04:20:46
%S 1,2,1,2,3,2,2,4,2,5,6,2,3,8,8,2,8,14,10,2,5,16,20,12,2,13,30,30,14,2,
%T 8,30,48,40,16,2,21,60,78,54,18,2,13,56,106,112,68,20,2,34,116,184,
%U 166,86,22,2,21,102,224,286,224,104,24,2,55,218,408,452,310,126,26,2
%N Triangle read by rows: T(n,k) is the number of 00-avoiding binary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=floor(n/2)).
%C The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
%C A sequence is palindromic if and only if its degree of asymmetry is 0.
%C Number of entries in row n is 1+floor(n/2).
%C Sum of entries in row n is A000045(n+2) (Fibonacci).
%C T(n,0) = A053602(n+2) (= number of palindromic 00-avoiding binary words of length n).
%C Sum(k*T(n,k), k>=0) = A275436(n).
%F G.f.: G(t,z) = (1 + 2z + tz^2 +z^3 -tz^5)/(1 - z^2 - tz^2 - z^4 - tz^4 + tz^6).
%e Row 3 is [3,2] because the 00-avoiding binary words of length 3 are 010, 011, 101, 110, 111, having asymmetry degrees 0, 1, 0, 1, 0, respectively.
%e Triangle starts:
%e 1;
%e 2;
%e 1,2;
%e 3,2;
%e 2,4,2;
%e 5,6,2.
%p G := (1+2*z+t*z^2+z^3-t*z^5)/(1-z^2-t*z^2-z^4-t*z^4+t*z^6): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. (1/2)*n) end do; # yields sequence in triangular form
%t Table[BinCounts[#, {0, Floor[n/2] + 1, 1}] &@ Map[Total@ BitXor[Take[#, Ceiling[Length[#]/2]], Reverse@ Take[#, -Ceiling[Length[#]/2]]] &, Select[PadLeft[IntegerDigits[#, 2], n] & /@ Range[0, 2^n - 1], Length@ SequenceCases[#, {0, 0}] == 0 &]], {n, 0, 15}] // Flatten (* _Michael De Vlieger_, Aug 15 2016, Version 10.1 *)
%Y Cf. A000045, A053602, A275436
%K nonn,tabf
%O 0,2
%A _Emeric Deutsch_, Aug 15 2016
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