A276059 Sum of the asymmetry degrees of all compositions of n with parts in {3,4,5,6, ...}.

0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 14, 24, 38, 62, 98, 156, 242, 376, 580, 896, 1380, 2126, 3266, 5008, 7658, 11688, 17804, 27084, 41148, 62448, 94668, 143360, 216864, 327726, 494790, 746368, 1124950, 1694286, 2549942, 3835120, 5764274, 8658442, 12997998, 19501468

Offset: 0

Author

Emeric Deutsch, Aug 22 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(7) = 2 because the compositions of 7 with parts in {3,4,5,...} are 7, 34, and 43 and the sum of their asymmetry degrees is 0+1+1.
		

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

Crossrefs

Cf. A276058.

Programs

Maple

g := 2*z^7*(1-z)/((1+z)*(1-z+z^3)*(1-z-z^3)^2): gser := series(g, z = 0, 55): seq(coeff(gser, z, n), n = 0 .. 50);

Mathematica

CoefficientList[Series[2 x^7*(1 - x)/((1 + x) (1 - x + x^3) (1 - x - x^3)^2), {x, 0, 45}], x] (* Michael De Vlieger, Aug 28 2016 *)
LinearRecurrence[{2,0,-1,0,-1,2,0,-1,-1,-1},{0,0,0,0,0,0,0,2,2,4},50] (* Harvey P. Dale, Sep 11 2019 *)

PARI

concat(vector(7), Vec(2*x^7*(1-x)/((1+x)*(1-x+x^3)*(1-x-x^3)^2) + O(x^20))) \\ Colin Barker, Aug 28 2016

Formula

G.f. g(z) = 2*z^7*(1-z)/((1+z)*(1-z+z^3)*(1-z-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*A276058(n,k).