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A275437
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Triangle read by rows: T(n,k) is the number of 01-avoiding binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= floor(n/2)).
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4
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1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1
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OFFSET
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0,2
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COMMENTS
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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).
A sequence is palindromic if and only if its degree of asymmetry is 0.
Number of entries in row n is 1 + floor(n/2).
Sum of entries in row n is n+1.
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LINKS
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FORMULA
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T(2k,k)=1 (k >= 0); T(n,k)=2 if k <= floor(n/2); T(n,k)=0 if k > floor(n/2).
G.f.: G(t,z) = (1 + z)/((1 - z)(1 - tz^2)).
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EXAMPLE
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Row 4 is [2,2,1] because the 01-avoiding binary words of length 4 are 0000, 1000, 1100, 1110, and 1111, having asymmetry degrees 0, 1, 2, 1, and 0, respectively.
Triangle starts:
1;
2;
2, 1;
2, 2;
2, 2, 1;
2, 2, 2.
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MAPLE
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T:= proc(n, k) if n = 2*k then 1 elif k <= floor((1/2)*n) then 2 else 0 end if end proc: for n from 0 to 20 do seq(T(n, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
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MATHEMATICA
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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, 1}] == 0 &]], {n, 0, 15}] // Flatten (* Michael De Vlieger, Aug 15 2016, Version 10.1 *)
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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