A275434 Sum of the degrees of asymmetry of all compositions of n.

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

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.

Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

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 * A364316 A381123 A151258

Adjacent sequences: A275431 A275432 A275433 * A275435 A275436 A275437

Keyword

nonn,easy

Author

Emeric Deutsch, Jul 29 2016

Status

approved