OFFSET
0,4
COMMENTS
LINKS
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
FORMULA
T(n,k) = sum(2^j*binomial(k+j,k)*binomial(n-2-j,k-2), j=0..n-k).
G.f.: G(t,x) = (1-x)^2/(1-3*x+2*x^2-t*x).
The g.f. of column k is x^k/((1-2*x)^(k+1)*(1-x)^(k-1)) (we have a Riordan array).
T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), with T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=2, T(2,1)=4, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham, Nov 26 2013
EXAMPLE
T(2,1)=4 because we have (1/1), (2/0), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
2,4,1;
4,12,7,1;
8,32,31,10,1;
16,80,111,59,13,1;
MAPLE
T := proc (n, k) options operator, arrow: sum(2^j*binomial(k+j, k)*binomial(n-j-2, k-2), j = 0 .. n-k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 13 2010
STATUS
approved