A120910 Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.

3, 6, 3, 12, 12, 3, 24, 36, 18, 3, 48, 96, 72, 24, 3, 96, 240, 240, 120, 30, 3, 192, 576, 720, 480, 180, 36, 3, 384, 1344, 2016, 1680, 840, 252, 42, 3, 768, 3072, 5376, 5376, 3360, 1344, 336, 48, 3, 1536, 6912, 13824, 16128, 12096, 6048, 2016, 432, 54, 3, 3072

Offset: 1

Author

Emeric Deutsch, Jul 15 2006

Keywords

• nonn
• tabl

Comments

Row sums are the powers of 3 (A000244).

T(n,k) = 3*A038207(n-1,k).

T(n,k) = A120909(n,n-k).

Sum_{k>=0} k*T(n,k) = (n-1)*3^(n-1) = A036290(n-1).

Examples

			T(3,1)=12 because we have 001,002,011,022,100,110,112,122,200,211,220 and 221.
Triangle starts:
   3;
   6,  3
  12, 12,  3;
  24, 36, 18,  3;
  48, 96, 72, 24,  3;
		

Crossrefs

Cf. A000244, A038207, A120909, A036290.

Programs

Maple

T:=(n,k)->3*2^(n-k-1)*binomial(n-1,k): for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

Mathematica

sol=Solve[{a==v(z^2+a z),b==v(z^2+b z),c==v(z^2+c z)},{a,b,c}];f[z_,u_]:=1/(1-3z-a-b-c)/.sol/.v->u-1;nn=10;Drop[Map[Select[#,#>0&]&,Level[CoefficientList[Series[f[z,u],{z,0,nn}],{z,u}],{2}]],1]//Grid (* Geoffrey Critzer, May 19 2014 *)

Formula

T(n,k) = 3*2^(n-k-1)*binomial(n-1,k).

G.f.: (1 - (y - 1)*x)/(1 - (y + 2)*x). Generally for the number of length n words with k levels on an m-ary alphabet (m>1): (1 - (y - 1)*x)/(1 - (y + m - 1)*x). - Geoffrey Critzer, May 19 2014