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A275440
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Triangle read by rows: T(n,k) is the number of compositions of n into odd parts, having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).
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1
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1, 1, 1, 2, 1, 2, 3, 2, 2, 6, 5, 8, 3, 14, 4, 8, 22, 4, 5, 30, 20, 13, 52, 24, 8, 60, 68, 8, 21, 112, 92, 8, 13, 116, 192, 56, 34, 228, 284, 64, 21, 218, 484, 248, 16, 55, 446, 768, 312, 16, 34, 402, 1132, 872, 144, 89, 848, 1900, 1184, 160, 55, 730
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OFFSET
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0,4
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COMMENTS
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The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
Number of entries in row n is 1 + floor(n/4).
Sum of entries in row n is A000045(n) (Fibonacci).
T(n,0) = A053602(n) (= number of palindromic compositions of n into odd parts).
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REFERENCES
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S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
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LINKS
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FORMULA
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G.f.: G(t,z) = (1-z^4)(1+z-z^2)/(1-2z^2-2tz^4+z^6). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, we have G(t,z) =(1 + F(z))/(1 - F(z^2) - t(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.
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EXAMPLE
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Row 5 is [3,2] because the compositions of 5 into odd parts are 5, 113, 131, 311, and 11111, having asymmetry degrees 0, 1, 0, 1, 0, respectively.
Triangle starts:
1;
1;
1;
2;
1,2;
3,2;
2,6;
5,8; .
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MAPLE
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G := (1-z^4)*(1+z-z^2)/(1-2*z^2-2*t*z^4+z^6): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
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MATHEMATICA
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Table[BinCounts[#, {0, 1 + Floor[n/4], 1}] &@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[(# - 1)/2, Ceiling[Length[#]/2]], Reverse@ Take[(# - 1)/2, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; EvenQ@ a]], 1]]], {n, 0, 20}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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