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 A276065 Sum of the asymmetry degrees of all compositions of n with parts in {1,5}. 2
 0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 8, 16, 20, 34, 40, 64, 80, 130, 164, 256, 320, 490, 620, 944, 1200, 1800, 2290, 3400, 4344, 6406, 8206, 12008, 15408, 22404, 28810, 41672, 53680, 77258, 99662, 142808, 184480, 263320, 340578, 484392, 627200, 889160, 1152480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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. REFERENCES S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239. V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356. Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1). FORMULA G.f.: g(z) = 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)(1-z^2+z^3)). In the more general situation of compositions into a[1]=1}, we have  g(z) = (F(z)^2 - F(z^2))/((1+F(z))(1-F(z))^2). a(n) = Sum(k*A276064(n,k), k>=0). EXAMPLE a(8) = 4 because the compositions of 8 with parts in {1,5} are  5111, 1511, 1151, 1115, and 11111111, and the sum of their asymmetry degrees is 1+1+1+1+0. MAPLE g := 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)*(1-z^2+z^3)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40); MATHEMATICA 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 == 5]]], 1]]], {n, 0, 42}] // Flatten (* Michael De Vlieger, Aug 22 2016 *) PROG (PARI) concat(vector(6), Vec(2*x^6/((1-x+x^2)^2*(1-x^2-x^3)^2*(1+x+x^2)*(1-x^2+x^3)) + O(x^50))) \\ Colin Barker, Aug 28 2016 CROSSREFS Cf. A276064. Sequence in context: A094858 A029940 A045674 * A143483 A323093 A264788 Adjacent sequences:  A276062 A276063 A276064 * A276066 A276067 A276068 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Aug 22 2016 STATUS approved

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Last modified March 25 21:57 EDT 2019. Contains 321477 sequences. (Running on oeis4.)