A181371 Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).

1, 3, 8, 1, 21, 6, 55, 25, 1, 144, 90, 9, 377, 300, 51, 1, 987, 954, 234, 12, 2584, 2939, 951, 86, 1, 6765, 8850, 3573, 480, 15, 17711, 26195, 12707, 2305, 130, 1, 46368, 76500, 43398, 10008, 855, 18, 121393, 221016, 143682, 40426, 4740, 183, 1, 317811

Offset

0,2

Comments

Row n contains 1 + floor(n/2) entries.

Sum of entries in row n is 3^n = A000244(n).

T(n,0) = F(2n+2) = A001906(n+1) (even-subscripted Fibonacci numbers).

T(n,1) = A001871(n-2).

Sum_{k>=0}k*T(n,k) = (n-1)*3^(n-2) = A027471(n) (n>=1).

Links

Alois P. Heinz, Rows n = 0..200, flattened

Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.

Formula

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

Example

T(3,1)=6 because we have 010, 011, 012, 001, 101 and 201.
T(4,2)=1 because we have 0101.
Triangle starts:
    1;
    3;
    8,  1;
   21,  6;
   55, 25,  1;
  144, 90,  9;

Maple

G := 1/(1-3*z+z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000244, A001871, A001906, A027471.

Sequence in context: A197725 A288875 A152230 * A118357 A278866 A281287

Adjacent sequences: A181368 A181369 A181370 * A181372 A181373 A181374

Keyword

nonn,tabf

Author

Emeric Deutsch, Oct 31 2010

Status

approved