login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A275445 Sum of the asymmetry degrees of all compositions of n with parts in {1,2,3}. 2
0, 0, 0, 2, 4, 10, 22, 50, 106, 222, 458, 936, 1890, 3788, 7540, 14924, 29388, 57620, 112540, 219062, 425112, 822726, 1588314, 3059470, 5881254, 11284514, 21614774, 41336300, 78936358, 150533496, 286708744, 545428024, 1036468344, 1967555208, 3731449176, 7070218506, 13384916364, 25319020898, 47857031870, 90391975562, 170614347714 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 (1,2,2,0,-4,-6,-6,-3,-1).

FORMULA

G.f. g(z) = 2*z^3*(1+z+z^2)/((1+z)(1+z^2)(1-z-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*A275444(n,k).

EXAMPLE

a(4) = 4 because the compositions of 4 with parts in {1,2,3} are  13, 31, 22, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 1 + 1 + 0 + 1 + 0 + 1 + 0 = 4.

MAPLE

g := 2*z^3*(1+z+z^2)/((1+z)*(1+z^2)*(1-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]], ReverseTake[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 3]], 1]]], {n, 0, 24}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

PROG

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

CROSSREFS

Cf. A275444.

Sequence in context: A239075 A260916 A192627 * A075560 A078040 A240041

Adjacent sequences:  A275442 A275443 A275444 * A275446 A275447 A275448

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Aug 17 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)