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

Author

Emeric Deutsch, Aug 17 2016

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A276053, A276055.

Programs

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 *)

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

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-2*z^2+z^3)^2). In the more general situation of compositions into a[1]=1} z^(a[j]), 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).