A134426 Triangle read by rows: T(n,k) is the number of paths of length n in the first quadrant, starting at the origin, ending at height k and consisting of 2 kind of upsteps U=(1,1) (U1 and U2), 3 kind of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1).
1, 3, 2, 11, 12, 4, 45, 62, 36, 8, 197, 312, 240, 96, 16, 903, 1570, 1440, 784, 240, 32, 4279, 7956, 8244, 5472, 2320, 576, 64, 20793, 40670, 46116, 35224, 18480, 6432, 1344, 128, 103049, 209712, 254912, 216384, 132320, 57600, 17024, 3072, 256
Offset: 0
Examples
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;
Programs
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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
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).
Comments