login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A275434 Sum of the degrees of asymmetry of all compositions of n. 1
0, 0, 0, 2, 4, 12, 28, 68, 156, 356, 796, 1764, 3868, 8420, 18204, 39140, 83740, 178404, 378652, 800996, 1689372, 3553508, 7456540, 15612132, 32622364, 68040932, 141674268, 294533348, 611436316, 1267611876, 2624702236, 5428361444, 11214636828 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry 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 degree of asymmetry is 0.

LINKS

Table of n, a(n) for n=0..32.

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

FORMULA

G.f.: g(z) = 2z^3(1-z)/((1-2z)(1-z-2z^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) = - (4/9)*(-1)^n + (3n - 2)*2^n/36  for n>=2; a(1)=0

a(n) = Sum(k*A275433(n,k), k>=0).

a(n) = 2*A059570(n-2) for n>=3. - Alois P. Heinz, Jul 29 2016

EXAMPLE

a(4) = 4 because the compositions 4, 13, 22, 31, 112, 121, 211, 1111 have degrees of asymmetry 0, 1, 0, 1, 1, 0, 1, 0, respectively.

MAPLE

g := 2*z^3*(1-z)/((1-2*z)*(1-z-2*z^2)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

a := proc(n) if n = 0 then 0 elif n = 1 then 0 else -(4/9)*(-1)^n+(1/36)*(3*n-2)*2^n end if end proc: seq(a(n), n = 0 .. 32);

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[b[n - j, If[i==0, j, 0]] If[i > 0 && i != j, x, 1], {j, 1, n}]]];

a[n_] := Function[p, Sum[i Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0]];

a /@ Range[0, 32] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz in A275433 *)

CROSSREFS

Cf. A059570, A275433.

Sequence in context: A292065 A193893 A096581 * A151258 A148175 A148176

Adjacent sequences:  A275431 A275432 A275433 * A275435 A275436 A275437

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 29 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 25 17:05 EDT 2021. Contains 348255 sequences. (Running on oeis4.)