A274496 Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).

1, 2, 2, 2, 4, 4, 4, 8, 4, 8, 16, 8, 8, 24, 24, 8, 16, 48, 48, 16, 16, 64, 96, 64, 16, 32, 128, 192, 128, 32, 32, 160, 320, 320, 160, 32, 64, 320, 640, 640, 320, 64, 64, 384, 960, 1280, 960, 384, 64, 128, 768, 1920, 2560, 1920, 768, 128

Offset: 0

Author

Emeric Deutsch, Jul 27 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) = A274497(n).

Examples

			From _Andrew Howroyd_, Jan 10 2018: (Start)
Triangle begins:
   1;
   2;
   2,   2;
   4,   4;
   4,   8,   4;
   8,  16,   8;
   8,  24,  24,   8;
  16,  48,  48,  16;
  16,  64,  96,  64,  16;
  32, 128, 192, 128,  32;
  32, 160, 320, 320, 160, 32;
  ...
(End)
T(4,0) = 4 because we have 0000, 0110, 1001, and 1111.
T(4,1) = 8 because we have 0001, 0010, 0100, 1000, 0111, 1011, 1101, and 1110.
T(4,2) = 4 because we have 0011, 0101, 1010, and 1100.
		

Crossrefs

Cf. A274497, A274498, A274499.

Programs

Maple

T := proc(n, k) options operator, arrow: 2^ceil((1/2)*n)*binomial(floor((1/2)*n), k) end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Mathematica

Table[2^Ceiling[n/2] Binomial[Floor[n/2], k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Michael De Vlieger, Jan 11 2018 *)

PARI

T(n,k) = 2^ceil(n/2)*binomial(floor(n/2), k);
for(n=0, 10, for(k=0, n\2, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Jan 10 2018

Formula

T(n,k) = 2^ceiling(n/2)*binomial(floor(n/2),k).

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

The row generating polynomials P[n] satisfy P[n] = 2(1 + t)P[n-2] (n >= 2). Easy to see if we note that the binary words of length n (n >= 2) are 0w0, 0w1, 1w0, and 1w1, where w is a binary word of length n-2.