A275443 Sum of the asymmetry degrees of all compositions of n without 2's.

0, 0, 0, 0, 2, 4, 8, 16, 34, 68, 134, 260, 502, 960, 1824, 3444, 6472, 12108, 22566, 41912, 77608, 143312, 263990, 485196, 889938, 1629256, 2977642, 5433344, 9899776, 18013288, 32734928, 59417944, 107732106, 195130092, 353087560, 638329168, 1153012298

Offset

0,5

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

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

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

Formula

G.f.: g(z) = 2z^4*(1-z)/((1+z)(1-2z+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*A275442(n,k).

Example

a(5) = 4 because the compositions of 5 without 2's are 5, 41, 14, 311, 131, 113, and 11111 and the sum of their asymmetry degrees is 0+1+1+1+0+1+0=4.

Maple

g := 2*z^4*(1-z)/((1+z)*(1-2*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]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; a == 2]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

Prog

(PARI) concat(vector(4), Vec(2*x^4*(1-x)/((1+x)*(1-2*x+x^2-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 29 2016

Crossrefs

Cf. A180177, A275442.

Sequence in context: A210541 A275545 A273972 * A288170 A088325 A215930

Adjacent sequences: A275440 A275441 A275442 * A275444 A275445 A275446

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 16 2016

Status

approved