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A275439 Sum of the asymmetry degrees of all compositions of n with parts in {1,2}.

%I A275439 #26 May 20 2025 21:40:28
%S A275439 0,0,0,2,2,6,12,22,42,78,140,252,448,788,1380,2402,4158,7170,12316,
%T A275439 21082,35982,61246,103992,176184,297888,502728,846984,1424738,2393114,
%U A275439 4014270,6725196,11253694,18810930,31410894,52400132,87335604,145438624,242001692
%N A275439 Sum of the asymmetry degrees of all compositions of n with parts in {1,2}.
%C A275439 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 A275439 A sequence is palindromic if and only if its asymmetry degree is 0.
%H A275439 Colin Barker, <a href="/A275439/b275439.txt">Table of n, a(n) for n = 0..1000</a>
%H A275439 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1,-2,-3,-1).
%F A275439 G.f.: g(z) = 2*z^3/((1+z+z^2)*(1-z-z^2)^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 A275439 a(n) = (n+1)/2-(3/2)*floor((n+2)/3)+(3/5)*(n+1)*f(n)-(1/10)*(2*n+5)*f(n+1), where f(j) = A000045(j) are the Fibonacci numbers.
%F A275439 a(n) = Sum_{k>=0} k*A275438(n,k).
%F A275439 a(n) = 2*A291408(n-4) for n>=4. - _Alois P. Heinz_, May 20 2025
%e A275439 a(5) = 6 because the compositions of 5 with parts in {1,2} are 122, 212, 221, 1112, 1121, 1211, 2111, and 11111 and the sum of their asymmetry degrees is 1 + 0 + 1 + 1 + 1 + 1 + 1 + 0 = 6.
%p A275439 f := n -> combinat:-fibonacci(n):
%p A275439 a := n -> (n+1)/2-(3/2)*floor((n+2)/3)+(3/5)*(n+1)*f(n)-(1/10)*(2*n+5)*f(n+1):
%p A275439 seq(a(n), n = 0..40);
%p A275439 # alternative program:
%p A275439 g := 2*z^3/((1+z+z^2)*(1-z-z^2)^2):
%p A275439 gser := series(g, z=0, 45):
%p A275439 seq(coeff(gser, z, n), n = 0..40);
%t A275439 Join[{0}, Table[Total@ Map[Total, Map[BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]] &,
%t A275439 Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 2]], 1]]], {n, 30}]] (* _Michael De Vlieger_, Aug 17 2016 *)
%o A275439 (PARI) concat(vector(3), Vec(2*x^3/((1+x+x^2)*(1-x-x^2)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016
%Y A275439 Cf. A000045, A275438, A291408.
%K A275439 nonn,easy
%O A275439 0,4
%A A275439 _Emeric Deutsch_, Aug 16 2016