OFFSET
1,1
COMMENTS
REFERENCES
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
FORMULA
G.f. for 2-compositions with all entries <= k is h(k,z)=(1-z)^2/(1-4z+2z^2+2z^{k+1}-z^{2k+2}).
G.f. for 2-compositions with largest entry k is f(k,z)=h(k,z)-h(k-1,z) (these are the column g.f.'s).
G.f. = G(t,z)=Sum(f(k,z)*t^k, k=1..infinity).
EXAMPLE
T(3,3)=2 because we have (0/3) and (3/0) (the 2-compositions are written as (top row/bottom row).
Triangle starts:
2;
5,2;
12,10,2;
29,41,10,2;
70,152,46,10,2;
MAPLE
h := proc (k) options operator, arrow: (1-z)^2/(1-4*z+2*z^2+2*z^(k+1)-z^(2*k+2)) end proc: f := proc (k) options operator, arrow; simplify(h(k)-h(k-1)) end proc: G := sum(f(k)*t^k, k = 1 .. 30): Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 11 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 15 2010
STATUS
approved