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 A275443 Sum of the asymmetry degrees of all compositions of n without 2's. 2
 0, 0, 0, 0, 2, 4, 8, 16, 34, 68, 134, 260, 502, 960, 1824, 3444, 6472, 12108, 22566, 41912, 77608, 143312, 263990, 485196, 889938, 1629256, 2977642, 5433344, 9899776, 18013288, 32734928, 59417944, 107732106, 195130092, 353087560, 638329168, 1153012298 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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 P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003. 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 (3,-2,0,1,-3,1,-1). FORMULA G.f.: g(z) = 2z^4*(1-z)/((1+z)(1-2z+z^2-z^3)^2). 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 >= 0} k*A275442(n,k). EXAMPLE a(5) = 4 because the compositions of 5 without 2's are 5, 41, 14, 311, 131, 113, and 11111 and the sum of their asymmetry degrees is 0+1+1+1+0+1+0=4. MAPLE g := 2*z^4*(1-z)/((1+z)*(1-2*z+z^2-z^3)^2): 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_, ___} /; a == 2]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 17 2016 *) PROG (PARI) concat(vector(4), Vec(2*x^4*(1-x)/((1+x)*(1-2*x+x^2-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 29 2016 CROSSREFS Cf. A180177, A275442. Sequence in context: A210541 A275545 A273972 * A288170 A088325 A215930 Adjacent sequences: A275440 A275441 A275442 * A275444 A275445 A275446 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Aug 16 2016 STATUS approved

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Last modified September 16 23:59 EDT 2024. Contains 375984 sequences. (Running on oeis4.)