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

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) = 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.
		

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).

Crossrefs

Cf. A275446.

Programs

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 *)

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

Formula

G.f.: g(z) = 2*z^3*(1-z^2)/((1+z^2)*(1-z-2*z^2+z^4)^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*A275446(n,k).