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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
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
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 31 2010
STATUS
approved