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

%I A275434 #21 Apr 11 2025 06:58:32
%S A275434 0,0,0,2,4,12,28,68,156,356,796,1764,3868,8420,18204,39140,83740,
%T A275434 178404,378652,800996,1689372,3553508,7456540,15612132,32622364,
%U A275434 68040932,141674268,294533348,611436316,1267611876,2624702236,5428361444,11214636828
%N A275434 Sum of the degrees of asymmetry of all compositions of n.
%C A275434 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 A275434 A sequence is palindromic if and only if its degree of asymmetry is 0.
%H A275434 V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%H A275434 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4).
%F A275434 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]<a[2]<a[3]<..., denoting F(z) = Sum_{j>=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).
%F A275434 a(n) = -(4/9)*(-1)^n + (3*n - 2)*2^n/36 for n>=2; a(0) = a(1) = 0.
%F A275434 a(n) = Sum_{k>=0} k*A275433(n,k).
%F A275434 a(n) = 2*A059570(n-2) for n>=3. - _Alois P. Heinz_, Jul 29 2016
%e A275434 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 A275434 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 A275434 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 A275434 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 A275434 a[n_] := Function[p, Sum[i Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0]];
%t A275434 a /@ Range[0, 32] (* _Jean-François Alcover_, Nov 24 2020, after _Alois P. Heinz_ in A275433 *)
%Y A275434 Cf. A059570, A275433.
%K A275434 nonn,easy
%O A275434 0,4
%A A275434 _Emeric Deutsch_, Jul 29 2016