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Sum of the degrees of asymmetry of all compositions of n.
1

%I #15 Jul 22 2022 10:23:56

%S 0,0,0,2,4,12,28,68,156,356,796,1764,3868,8420,18204,39140,83740,

%T 178404,378652,800996,1689372,3553508,7456540,15612132,32622364,

%U 68040932,141674268,294533348,611436316,1267611876,2624702236,5428361444,11214636828

%N Sum of the degrees of asymmetry of all compositions of n.

%C 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).

%C A sequence is palindromic if and only if its degree of asymmetry is 0.

%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4).

%F 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).

%F a(n) = - (4/9)*(-1)^n + (3n - 2)*2^n/36 for n>=2; a(1)=0

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

%F a(n) = 2*A059570(n-2) for n>=3. - _Alois P. Heinz_, Jul 29 2016

%e 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.

%p 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);

%p 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);

%t 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}]]];

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

%t a /@ Range[0, 32] (* _Jean-François Alcover_, Nov 24 2020, after _Alois P. Heinz_ in A275433 *)

%Y Cf. A059570, A275433.

%K nonn,easy

%O 0,4

%A _Emeric Deutsch_, Jul 29 2016