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A274498 Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2). 3
1, 3, 3, 6, 9, 18, 9, 36, 36, 27, 108, 108, 27, 162, 324, 216, 81, 486, 972, 648, 81, 648, 1944, 2592, 1296, 243, 1944, 5832, 7776, 3888, 243, 2430, 9720, 19440, 19440, 7776, 729, 7290, 29160, 58320, 58320, 23328, 729, 8748, 43740, 116640, 174960, 139968, 46656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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(kT(n,k),k>=0) = A274499(n).

LINKS

Table of n, a(n) for n=0..48.

FORMULA

T(n,k) = 2^k*3^ceiling(n/2)*binomial(floor(n/2),k).

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

The row generating polynomials P[n] satisfy P[n] = 3(1 + 2t)P[n-2] (n>=2). Easy to see if we note that the ternary words of length n (n>=2) are 0w0, 0w1, 0w2, 1w0, 1w1, 1w2, 2w0, 2w1, 2w2, where w is a ternary word of length n - 2.

EXAMPLE

From Andrew Howroyd, Jan 10 2018: (Start)

Triangle begins:

   1;

   3;

   3,   6;

   9,  18;

   9,  36,   36;

  27, 108,  108;

  27, 162,  324,  216;

  81, 486,  972,  648;

  81, 648, 1944, 2592, 1296;

  ...

(End)

T(2,0) = 3 because we have 00, 11, and 22.

T(2,1) = 6 because we have 01, 02, 10, 12, 20, and 21.

MAPLE

T := proc(n, k) options operator, arrow: 2^k*3^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

PROG

(PARI)

T(n, k) = 2^k*3^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

CROSSREFS

Cf. A274496, A274497, A274499.

Sequence in context: A097135 A293677 A293679 * A167786 A167787 A185957

Adjacent sequences:  A274495 A274496 A274497 * A274499 A274500 A274501

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 27 2016

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)