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A275447 Sum of the asymmetry degrees of all compositions of n with parts in {2,1,3,5,7,9,...}.

%I A275447 #21 Apr 12 2025 01:01:14
%S A275447 0,0,0,2,4,10,24,54,120,258,552,1164,2432,5042,10384,21268,43344,
%T A275447 87962,177840,358358,719964,1442584,2883504,5751020,11447164,22743262,
%U A275447 45110096,89334192,176658732,348875904,688122336,1355674528,2667921660,5245033102
%N A275447 Sum of the asymmetry degrees of all compositions of n with parts in {2,1,3,5,7,9,...}.
%C A275447 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 A275447 A sequence is palindromic if and only if its asymmetry degree is 0.
%D A275447 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A275447 Colin Barker, <a href="/A275447/b275447.txt">Table of n, a(n) for n = 0..1000</a>
%H A275447 Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.
%H A275447 V. E. Hoggatt, Jr. and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%H A275447 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-3,-2,-2,2,3,0,-1).
%F A275447 G.f.: g(z) = 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+z^4)^2). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum_{j>=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).
%F A275447 a(n) = Sum_{k>=0} k*A275446(n,k).
%e A275447 a(4) = 4 because the compositions of 4 with parts in {2,1,3,5,7,...} are 22, 31, 13, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 0 + 1 + 1 + 1 + 0 + 1 + 0 = 4.
%p A275447 g := 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+z^4)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
%t A275447 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_, ___} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 21}] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)
%o A275447 (PARI) concat(vector(3), Vec(2*x^3*(1-x^2)/((1+x^2)*(1-x-2*x^2+x^4)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016
%Y A275447 Cf. A275446.
%K A275447 nonn,easy
%O A275447 0,4
%A A275447 _Emeric Deutsch_, Aug 17 2016