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A276063
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Sum of the asymmetry degrees of all compositions of n with parts in {1,4}.
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2
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0, 0, 0, 0, 0, 2, 2, 4, 4, 8, 14, 20, 32, 44, 70, 104, 152, 228, 326, 488, 704, 1026, 1492, 2144, 3120, 4470, 6450, 9256, 13256, 19026, 27144, 38840, 55360, 78910, 112406, 159768, 227240, 322500, 457734, 648996, 919372, 1302114, 1842036, 2605452, 3682112
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OFFSET
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0,6
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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).
A sequence is palindromic if and only if its asymmetry degree is 0.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,2,-3,0,1,-3,0,0,-1).
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FORMULA
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G.f. g(z) = 2*z^5/((1+z+z^4)(1-z-z^4)^2). 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(z) = (F(z)^2 - F(z^2))/((1+F(z))(1-F(z))^2).
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EXAMPLE
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a(6) = 2 because the compositions of 6 with parts in {1,4} are 411,141,114, and 111111 and the sum of their asymmetry degrees is 1+0+1+0.
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MAPLE
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g := 2*z^5/((1+z+z^4)*(1-z-z^4)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
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MATHEMATICA
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Table[Total@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; Nor[a == 1, a == 4]]], 1]]], {n, 0, 38}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
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PROG
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(PARI) concat(vector(5), Vec(2*x^5/((1+x+x^4)*(1-x-x^4)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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