A276054 Sum of the asymmetry degrees of all compositions of n with parts in {1,2,4,6,8,10,...}.

0, 0, 0, 2, 2, 8, 16, 34, 72, 146, 294, 590, 1156, 2278, 4422, 8572, 16510, 31682, 60558, 115398, 219190, 415348, 784996, 1480600, 2786818, 5236078, 9821222, 18393268, 34397388, 64241880, 119831316, 223266154, 415532226, 772587316, 1435082052, 2663283782

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 (1,4,-1,-5,-3,-1,0,5,2,-4,1).

Formula

G.f.: g(z)=2*z^3*(1-z^2)(1+z^3-z^4)/((1+z^2)(1+z-z^3)(1-z-2z^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*A276053(n,k).

Example

a(4) = 2 because the compositions of 4 with parts in {1,2,4,6,8,...} are  4, 22, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 0 + 0 + 1 + 0 + 1 + 0 = 2.

Maple

g:= 2*z^3*(1-z^2)*(1+z^3-z^4)/((1+z^2)*(1+z-z^3)*(1-z-2*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]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; Nor[a == 1, EvenQ@ a]]], 1]]], {n, 0, 23}] // Flatten (* or *)
CoefficientList[Series[2 x^3*(1 - x^2) (1 + x^3 - x^4)/((1 + x^2) (1 + x - x^3) (1 - x - 2 x^2 + x^3)^2), {x, 0, 35}], x] (* Michael De Vlieger, Aug 28 2016 *)

Prog

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

Crossrefs

Cf. A276053, A276055.

Sequence in context: A342835 A361294 A220172 * A192305 A228797 A052970

Adjacent sequences: A276051 A276052 A276053 * A276055 A276056 A276057

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 17 2016

Status

approved