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A276057 Sum of the asymmetry degrees of all compositions of n with parts in {1,3}. 2
0, 0, 0, 0, 2, 2, 4, 6, 14, 18, 36, 50, 94, 130, 236, 330, 580, 816, 1404, 1984, 3354, 4758, 7932, 11286, 18600, 26532, 43308, 61908, 100232, 143540, 230776, 331008, 528950, 759726, 1207584, 1736534, 2747242, 3954826, 6230444, 8977686, 14090410, 20320854 (list; graph; refs; listen; history; text; internal format)
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

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

FORMULA

G.f.: g(z) = 2*z^4/((1+z+z^3)(1-z-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*A276056(n,k).

EXAMPLE

a(6) = 4  because the compositions of 6 with parts in {1,3} are  33, 3111, 1311, 1131, 1113, and 111111 and the sum of their asymmetry degrees is 0 + 1+1+1+1+0.

MAPLE

g:=2*z^4/((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_, ___} /; Nor[a == 1, a == 3]]], 1]]], {n, 0, 34}] // Flatten (* or *)

CoefficientList[Series[2 x^4/((1 + x + x^3) (1 - x - x^3)^2), {x, 0, 41}], x] (* Michael De Vlieger, Aug 28 2016 *)

PROG

(PARI) concat(vector(4), Vec(2*x^4/((1+x+x^3)*(1-x-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016

CROSSREFS

Cf. A276056.

Sequence in context: A181926 A061894 A116684 * A116637 A153961 A134041

Adjacent sequences:  A276054 A276055 A276056 * A276058 A276059 A276060

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Aug 18 2016

STATUS

approved

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Last modified February 20 15:52 EST 2020. Contains 332078 sequences. (Running on oeis4.)