A128235 Triangle read by rows: T(n,k) is the number of sequences of length n on the alphabet {0,1,2,3}, containing k subsequences 00 (0<=k<=n-1).

1, 4, 15, 1, 57, 6, 1, 216, 33, 6, 1, 819, 162, 36, 6, 1, 3105, 756, 189, 39, 6, 1, 11772, 3402, 945, 216, 42, 6, 1, 44631, 14931, 4536, 1143, 243, 45, 6, 1, 169209, 64314, 21168, 5778, 1350, 270, 48, 6, 1, 641520, 273051, 96633, 28323, 7128, 1566, 297, 51

Offset: 0

Author

Emeric Deutsch, Feb 27 2007

Keywords

• nonn
• tabf

Comments

Row n has n terms (n>=1). T(n,0) = A125145(n). Sum(k*T(n,k), k=0..n-1) = (n-1)*4^(n-2) = A002697(n-1).

Examples

			T(4,2) = 6 because we have 0001, 0002, 0003, 1000, 2000 and 3000.
Triangle starts:
1;
4;
15,    1;
57,    6,  1;
216,  33,  6, 1;
819, 162, 36, 6, 1;
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Cf. A125145, A002697.

Programs

Maple

G:=(1+z-t*z)/(1-3*z-3*z^2-t*z+3*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

Formula

G.f.: (1+z-tz)/(1-3z-3z^2-tz+3tz^2).