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A276065 Sum of the asymmetry degrees of all compositions of n with parts in {1,5}.

%I A276065 #20 Apr 12 2025 05:17:51
%S A276065 0,0,0,0,0,0,2,2,4,4,6,8,16,20,34,40,64,80,130,164,256,320,490,620,
%T A276065 944,1200,1800,2290,3400,4344,6406,8206,12008,15408,22404,28810,41672,
%U A276065 53680,77258,99662,142808,184480,263320,340578,484392,627200,889160,1152480
%N A276065 Sum of the asymmetry degrees of all compositions of n with parts in {1,5}.
%C A276065 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 A276065 A sequence is palindromic if and only if its asymmetry degree is 0.
%D A276065 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276065 Colin Barker, <a href="/A276065/b276065.txt">Table of n, a(n) for n = 0..1000</a>
%H A276065 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 A276065 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 A276065 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1).
%F A276065 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]<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 A276065 a(n) = Sum_{k>=0} k*A276064(n,k).
%e A276065 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.
%p A276065 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);
%t A276065 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 *)
%t A276065 LinearRecurrence[{1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1},{0,0,0,0,0,0,2,2,4,4,6,8,16,20,34},50] (* _Harvey P. Dale_, Aug 29 2021 *)
%o A276065 (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
%Y A276065 Cf. A276064.
%K A276065 nonn,easy
%O A276065 0,7
%A A276065 _Emeric Deutsch_, Aug 22 2016