A276063 - OEIS

%I #14 Aug 28 2016 11:07:02

%S 0,0,0,0,0,2,2,4,4,8,14,20,32,44,70,104,152,228,326,488,704,1026,1492,

%T 2144,3120,4470,6450,9256,13256,19026,27144,38840,55360,78910,112406,

%U 159768,227240,322500,457734,648996,919372,1302114,1842036,2605452,3682112

%N Sum of the asymmetry degrees of all compositions of n with parts in {1,4}.

%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="/A276063/b276063.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_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,2,-3,0,1,-3,0,0,-1).

%F G.f. g(z) = 2*z^5/((1+z+z^4)(1-z-z^4)^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*A276062(n,k), k>=0).

%e a(6) = 2 because the compositions of 6 with parts in {1,4} are 411,141,114, and 111111 and the sum of their asymmetry degrees is 1+0+1+0.

%p g := 2*z^5/((1+z+z^4)*(1-z-z^4)^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]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; Nor[a == 1, a == 4]]], 1]]], {n, 0, 38}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)

%o (PARI) concat(vector(5), Vec(2*x^5/((1+x+x^4)*(1-x-x^4)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016

%Y Cf. A276062.

%K nonn,easy

%O 0,6

%A _Emeric Deutsch_, Aug 22 2016