OFFSET
0,2
FORMULA
T(n,k) = ((k+1)*2^k/(n+1))*Sum_{j=0..n-k} binomial(n+1, j)*binomial(n+1, k+j+1)*2^j (0 <= k <= n).
G.f.: g/(1-2*t*z*g), where g = 1 + 3*z*g + 2*z^2*g^2 is the g.f. of the little Schroeder numbers 1, 3, 11, 45, 197, ... (A001003).
EXAMPLE
T(2,1)=12 because we have 6 paths of shape FU and 6 paths of shape UF.
Triangle starts:
1;
3, 2;
11, 12, 4;
45, 62, 36, 8;
197, 312, 240, 96, 16;
MAPLE
T:=proc(n, k) options operator, arrow: 2^k*(k+1)*(sum(2^j*binomial(n+1, j)*binomial(n+1, k+1+j), j=0..n-k))/(n+1) end proc: for n from 0 to 8 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 05 2007
STATUS
approved