A275439 - OEIS

%I #20 Aug 28 2016 13:01:42

%S 0,0,0,2,2,6,12,22,42,78,140,252,448,788,1380,2402,4158,7170,12316,

%T 21082,35982,61246,103992,176184,297888,502728,846984,1424738,2393114,

%U 4014270,6725196,11253694,18810930,31410894,52400132,87335604,145438624,242001692

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

%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.

%H Colin Barker, <a href="/A275439/b275439.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1,-2,-3,-1).

%F G.f.: g(z) = 2z^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(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) = (n+1)/2-(3/2)*floor((n+2)/3)+(3/5)*(n+1)f(n)-(1/10)*(2n+5)f(n+1), where f(j)= A000045(j) are the Fibonacci numbers.

%F a(n) = Sum_{k>=0} k*A275438(n,k).

%e 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 f := n -> combinat:-fibonacci(n):

%p 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 seq(a(n), n = 0..40);

%p # alternative program:

%p g := 2*z^3/((1+z+z^2)*(1-z-z^2)^2):

%p gser := series(g, z=0, 45):

%p seq(coeff(gser, z, n), n = 0..40);

%t Join[{0}, Table[Total@ Map[Total, Map[BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]] &,

%t Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 2]], 1]]], {n, 30}]] (* _Michael De Vlieger_, Aug 17 2016 *)

%o (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 Cf. A000045, A275438.

%K nonn,easy

%O 0,4

%A _Emeric Deutsch_, Aug 16 2016