%I #14 Aug 29 2016 11:15:05

%S 0,0,0,0,2,4,8,16,34,68,134,260,502,960,1824,3444,6472,12108,22566,

%T 41912,77608,143312,263990,485196,889938,1629256,2977642,5433344,

%U 9899776,18013288,32734928,59417944,107732106,195130092,353087560,638329168,1153012298

%N Sum of the asymmetry degrees of all compositions of n without 2's.

%C 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).

%C A sequence is palindromic if and only if its asymmetry degree is 0.

%D S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

%H Colin Barker, <a href="/A275443/b275443.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.

%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,1,-3,1,-1).

%F 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]<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).

%F a(n) = Sum_{k >= 0} k*A275442(n,k).

%e 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.

%p 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);

%t 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 *)

%o (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

%Y Cf. A180177, A275442.

%K nonn,easy

%O 0,5

%A _Emeric Deutsch_, Aug 16 2016