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

Author

Emeric Deutsch, Jul 29 2016

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A059570, A275433.

Programs

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 *)

Formula

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

a(n) = Sum_{k>=0} k*A275433(n,k).

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