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

0,8

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

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]<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*A276058(n,k), k>=0).

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A276058.

Sequence in context: A300415 A355908 A306731 * A034410 A192682 A050194

Adjacent sequences: A276056 A276057 A276058 * A276060 A276061 A276062

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 22 2016

Status

approved