OFFSET
0,6
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.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
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 (1,1,-1,1,2,-3,0,1,-3,0,0,-1).
FORMULA
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).
a(n) = Sum(k*A276062(n,k), k>=0).
EXAMPLE
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.
MAPLE
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);
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_, ___} /; Nor[a == 1, a == 4]]], 1]]], {n, 0, 38}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 22 2016
STATUS
approved