A132279 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k doublerises (i.e., UU's) (0 <= k <= floor(n/2) - 1 for n >= 2).
1, 1, 3, 6, 15, 1, 36, 4, 91, 17, 1, 232, 60, 5, 603, 206, 26, 1, 1585, 676, 110, 6, 4213, 2174, 444, 37, 1, 11298, 6868, 1687, 182, 7, 30537, 21446, 6196, 841, 50, 1, 83097, 66356, 22100, 3612, 280, 8, 227475, 203914, 77138, 14833, 1455, 65, 1
Offset: 0
Examples
Triangle starts:
1;
1;
3;
6;
15, 1;
36, 4;
91, 17, 1;
232, 60, 5;
T(5,1)=4 because we have UUhDD, UUDhD, hUUDD and UUDDh.
Programs
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Maple
G:=((1-z-2*z^2+z^2*t-sqrt((1+z-z^2*t)*(1-3*z-z^2*t)))*1/2)/(z^2*(t+z+z^2-z*t-z^2*t)): Gser:=simplify(series(G,z=0,18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser,z,n)) end do: 1; 1; for n from 2 to 14 do seq(coeff(P[n],t,j),j= 0..floor((1/2)*n)-1) end do; # yields sequence in triangular form
Formula
G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*G + z^2*(t(G-1-zG-z^2*G) + 1 + zG + z^2*G)G (see explicit expression at the Maple program).
G.f.: G = 2/(1-z-2*z^2+t*z^2+sqrt(1-2*z-3*z^2-2*t*z^2+2*t*z^3+t^2*z^4)). - Olivier Gérard, Sep 27 2007
Comments