A275438 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,2} having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).

1, 1, 2, 1, 2, 3, 2, 2, 6, 5, 4, 4, 3, 14, 4, 8, 10, 16, 5, 30, 12, 8, 13, 20, 48, 8, 8, 60, 36, 40, 21, 40, 124, 32, 16, 13, 116, 88, 144, 16, 34, 76, 292, 112, 96, 21, 218, 204, 432, 80, 32, 55, 142, 648, 320, 400, 32, 34, 402, 444, 1160, 320, 224

Offset: 0

Author

Emeric Deutsch, Aug 16 2016

Keywords

• nonn
• tabf

Comments

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

Number of entries in row n is 1 + floor(n/3).

Sum of entries in row n is A000045(n+1) (Fibonacci).

T(n,0) = A053602(n+1) (= number of palindromic compositions of n with parts in {1,2}).

Sum_{k>=0} k*T(n,k) = A275439(n).

Examples

			Row 4 is [3,2] because the compositions of 4 with parts in {1,2} are 22, 112, 121, 211, and 1111, having asymmetry degrees 0, 1, 0, 1, 0, respectively.
Triangle starts:
  1;
  1;
  2;
  1,2;
  3,2;
  2,6;
  5,4,4.
		

Links

• Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
• V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Crossrefs

Cf. A000045, A053602, A275439.

Programs

Maple

G:=(1+z+z^2)/(1-z^2-2*t*z^3-z^4): Gser:=simplify(series(G,z=0,25)): 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

Mathematica

Join[{{1}}, Table[BinCounts[#, {0, 1 + Floor[n/3], 1}] &@ Map[Total, Map[BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, _} /; a > 2]], 1]]], {n, 17}]] // Flatten (* Michael De Vlieger, Aug 17 2016 *)

Formula

G.f.: G(t,z) = (1+z+z^2)/(1-z^2-2*t*z^3-z^4). 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.