A275447 Sum of the asymmetry degrees of all compositions of n with parts in {2,1,3,5,7,9,...}.

0, 0, 0, 2, 4, 10, 24, 54, 120, 258, 552, 1164, 2432, 5042, 10384, 21268, 43344, 87962, 177840, 358358, 719964, 1442584, 2883504, 5751020, 11447164, 22743262, 45110096, 89334192, 176658732, 348875904, 688122336, 1355674528, 2667921660, 5245033102

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

Formula

G.f.: g(z) = 2*z^3*(1-z^2)/((1+z^2)(1-z-2z^2+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>=0} k*A275446(n,k).

Example

a(4) = 4 because the compositions of 4 with parts in {2,1,3,5,7,...} are  22, 31, 13, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 0 + 1 + 1 +1 + 0 +1 + 0 = 4.

Maple

g := 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+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_, ___} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 21}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

Prog

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

Crossrefs

Cf. A275446.

Sequence in context: A356695 A089484 A132732 * A095214 A002525 A159328

Adjacent sequences: A275444 A275445 A275446 * A275448 A275449 A275450

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 17 2016

Status

approved