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%I #15 Jan 13 2023 10:31:51
%S 0,0,0,2,4,10,22,50,106,222,458,936,1890,3788,7540,14924,29388,57620,
%T 112540,219062,425112,822726,1588314,3059470,5881254,11284514,
%U 21614774,41336300,78936358,150533496,286708744,545428024,1036468344,1967555208,3731449176,7070218506,13384916364,25319020898,47857031870,90391975562,170614347714
%N Sum of the asymmetry degrees of all compositions of n with parts in {1,2,3}.
%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="/A275445/b275445.txt">Table of n, a(n) for n = 0..1000</a>
%H Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.
%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://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_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,2,0,-4,-6,-6,-3,-1).
%F G.f. g(z) = 2*z^3*(1+z+z^2)/((1+z)(1+z^2)(1-z-z^2-z^3)^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*A275444(n,k).
%F 4*a(n) = (-1)^(n+1) +A057077(n+1) -2*A000073(n) +4*A073778(n+2). - _R. J. Mathar_, Jan 13 2023
%e a(4) = 4 because the compositions of 4 with parts in {1,2,3} are 13, 31, 22, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 1 + 1 + 0 + 1 + 0 + 1 + 0 = 4.
%p g := 2*z^3*(1+z+z^2)/((1+z)*(1+z^2)*(1-z-z^2-z^3)^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, Ceiling[Length[#]/2]], ReverseTake[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 3]], 1]]], {n, 0, 24}] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)
%o (PARI) concat(vector(3), Vec(2*x^3*(1+x+x^2)/((1+x)*(1+x^2)*(1-x-x^2-x^3)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016
%Y Cf. A275444.
%K nonn,easy
%O 0,4
%A _Emeric Deutsch_, Aug 17 2016
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