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A276061 Sum of the asymmetry degrees of all compositions of n into parts congruent to 1 mod 3.

%I A276061 #23 Apr 12 2025 09:43:24
%S A276061 0,0,0,0,0,2,2,4,6,10,18,28,46,74,114,184,286,448,700,1080,1676,2582,
%T A276061 3970,6104,9338,14288,21808,33224,50580,76844,116640,176832,267740,
%U A276061 405058,612110,924204,1394266,2101558,3165406,4764184,7165530,10770386,16178378
%N A276061 Sum of the asymmetry degrees of all compositions of n into parts congruent to 1 mod 3.
%C A276061 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 A276061 A sequence is palindromic if and only if its asymmetry degree is 0.
%D A276061 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276061 Colin Barker, <a href="/A276061/b276061.txt">Table of n, a(n) for n = 0..1000</a>
%H A276061 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 A276061 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 A276061 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,0,-1,-1,-1,-2,1,0,1).
%F A276061 G.f.: g(z) = 2*z^5*(1-z^3)/((1+z)*(1-z+z^2)*(1+z-z^3)*(1-z-z^3)^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 A276061 a(n) = Sum_{k>=0} k*A276060(n,k).
%e A276061 a(7) = 4 because the compositions of 7 with parts in {1,4,7,10,...} are 7, 4111, 1411, 1141, 1114, and 1111111, and the sum of their asymmetry degrees is 0+1+1+1+1+0.
%p A276061 g := 2*z^5*(1-z^3)/((1+z)*(1-z+z^2)*(1+z-z^3)*(1-z-z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
%t A276061 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_, ___} /; Mod[a, 3] != 1]], 1]]], {n, 0, 36}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)
%o A276061 (PARI) concat(vector(5), Vec(2*x^5*(1-x^3)/((1+x)*(1-x+x^2)*(1+x-x^3)*(1-x-x^3)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016
%Y A276061 Cf. A276060.
%K A276061 nonn,easy
%O A276061 0,6
%A A276061 _Emeric Deutsch_, Aug 22 2016