OFFSET
0,5
COMMENTS
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).
A sequence is palindromic if and only if its asymmetry degree is 0.
REFERENCES
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,1,-3,1,-1).
FORMULA
G.f.: g(z) = 2z^4*(1-z)/((1+z)(1-2z+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).
a(n) = Sum_{k >= 0} k*A275442(n,k).
EXAMPLE
a(5) = 4 because the compositions of 5 without 2's are 5, 41, 14, 311, 131, 113, and 11111 and the sum of their asymmetry degrees is 0+1+1+1+0+1+0=4.
MAPLE
g := 2*z^4*(1-z)/((1+z)*(1-2*z+z^2-z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
MATHEMATICA
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_, ___} /; a == 2]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)
PROG
(PARI) concat(vector(4), Vec(2*x^4*(1-x)/((1+x)*(1-2*x+x^2-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 29 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 16 2016
STATUS
approved