A276065 Sum of the asymmetry degrees of all compositions of n with parts in {1,5}.

0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 8, 16, 20, 34, 40, 64, 80, 130, 164, 256, 320, 490, 620, 944, 1200, 1800, 2290, 3400, 4344, 6406, 8206, 12008, 15408, 22404, 28810, 41672, 53680, 77258, 99662, 142808, 184480, 263320, 340578, 484392, 627200, 889160, 1152480

Offset

0,7

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

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,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1).

Formula

G.f.: g(z) = 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)(1-z^2+z^3)). 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*A276064(n,k), k>=0).

Example

a(8) = 4 because the compositions of 8 with parts in {1,5} are  5111, 1511, 1151, 1115, and 11111111, and the sum of their asymmetry degrees is 1+1+1+1+0.

Maple

g := 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)*(1-z^2+z^3)): 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 == 5]]], 1]]], {n, 0, 42}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
LinearRecurrence[{1, 1, -1, 0, 1, 2, -3, 0, 0, 1, -3, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 8, 16, 20, 34}, 50] (* Harvey P. Dale, Aug 29 2021 *)

Prog

(PARI) concat(vector(6), Vec(2*x^6/((1-x+x^2)^2*(1-x^2-x^3)^2*(1+x+x^2)*(1-x^2+x^3)) + O(x^50))) \\ Colin Barker, Aug 28 2016

Crossrefs

Cf. A276064.

Sequence in context: A327851 A029940 A045674 * A325253 A143483 A323093

Adjacent sequences: A276062 A276063 A276064 * A276066 A276067 A276068

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 22 2016

Status

approved