A275441 - OEIS

%I #18 Jan 05 2025 19:51:40

%S 0,0,0,0,2,2,6,8,22,30,70,100,220,320,668,988,1994,2982,5858,8840,

%T 17010,25850,48910,74760,139512,214272,395256,609528,1113362,1722890,

%U 3120510,4843400,8708110,13551510,24207958,37759468,67068244,104827712,185250068

%N Sum of the asymmetry degrees of all compositions of n into odd parts.

%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="/A275441/b275441.txt">Table of n, a(n) for n = 0..1000</a>

%H V. E. Hoggatt, Jr., and M. 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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,0,-2,-3,1,1).

%F G.f.: g(z)= 2z^4*(1-z^2)/((1+z^2)(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) = Sum_{k>=0} k*A275440(n,k).

%e a(6) = 6 because the compositions of 6 into odd parts are 15, 51, 33, 1113, 1131, 1311, 3111, 111111 and the sum of their asymmetry degrees is 1 + 1 + 0 +1 + 1 + 1 + 1 + 0 = 6.

%p g:= 2*z^4*(1-z^2)/((1+z^2)*(1+z-z^2)*(1-z-z^2)^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)/2, Ceiling[Length[#]/2]], Reverse@ Take[(# - 1)/2, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; EvenQ@ a]], 1]]], {n, 0, 30}] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)

%t LinearRecurrence[{1,3,-2,0,-2,-3,1,1},{0,0,0,0,2,2,6,8},40] (* _Harvey P. Dale_, Jan 13 2019 *)

%o (PARI) concat(vector(4), Vec(2*x^4*(1-x^2)/((1+x^2)*(1+x-x^2)*(1-x-x^2)^2) + O(x^50))) \\ _Colin Barker_, Aug 29 2016

%Y Cf. A275440.

%K nonn,easy

%O 0,5

%A _Emeric Deutsch_, Aug 16 2016