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

0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 18, 28, 46, 74, 114, 184, 286, 448, 700, 1080, 1676, 2582, 3970, 6104, 9338, 14288, 21808, 33224, 50580, 76844, 116640, 176832, 267740, 405058, 612110, 924204, 1394266, 2101558, 3165406, 4764184, 7165530, 10770386, 16178378

Offset: 0

Author

Emeric Deutsch, Aug 22 2016

Keywords

• nonn
• easy

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.

Examples

			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.
		

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
• 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,-1,0,-1,-1,-1,-2,1,0,1).

Crossrefs

Cf. A276060.

Programs

Maple

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);

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_, _} /; Mod[a, 3] != 1]], 1]]], {n, 0, 36}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

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

Formula

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]=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).

a(n) = Sum_{k>=0} k*A276060(n,k).