A275433 Triangle read by rows: T(n,k) is the number of compositions of n having degree of asymmetry equal to k (n>=0; 0<=k<=n/3).

1, 1, 2, 2, 2, 4, 4, 4, 12, 8, 20, 4, 8, 44, 12, 16, 68, 44, 16, 132, 100, 8, 32, 196, 252, 32, 32, 356, 500, 136, 64, 516, 1068, 384, 16, 64, 900, 1956, 1096, 80, 128, 1284, 3804, 2592, 384, 128, 2180, 6612, 6152, 1280, 32, 256, 3076, 12108, 13056, 4080, 192, 256, 5124, 20292, 27784, 11056, 1024, 512, 7172, 35644, 54816, 28960, 3904, 64

Offset: 0

Author

Emeric Deutsch, Jul 29 2016

Keywords

• nonn
• tabf

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.

Sum_{k>=0} k*T(n,k) = A275434(n).

Examples

			T(4,0) = 4 because we have 4, 22, 121, and 1111.
T(4,1) = 4 because we have 13, 31, 112, and 211.
Triangle starts:
  1;
  1;
  2;
  2,2;
  4,4;
  4,12;
  8,20,4.
		

Links

• Alois P. Heinz, Rows n = 0..250, flattened
• Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2002) no 1 #S2R7 eq. (1).
• V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Crossrefs

Cf. A275434.

Row sums give A011782.

Column k=0 gives A016116.

Programs

Maple

G := (1-z^2)/((1-z)*(1-2*z^2)-2*t*z^3): Gser := simplify(series(G, z = 0, 24)): for n from 0 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1, add(b(n-j,
      `if`(i=0, j, 0))*`if`(i>0 and i<>j, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 29 2016

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}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

G.f.: G(t,z) = (1-z^2)/((1-z)*(1-2*z^2) - 2*t*z^3). In the more general situation of compositions into a[1]=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.