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

0,3

Comments

Row n contains floor(n/2) terms (n>=2). Row sums yield A118720. T(n,0) = A005043(n+2) (the Riordan numbers).

Links

Table of n, a(n) for n=0..50.

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

Example

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.

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

Crossrefs

Cf. A005043, A118720.

Sequence in context: A324222 A058141 A144654 * A334042 A066107 A162276

Adjacent sequences: A132276 A132277 A132278 * A132280 A132281 A132282

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 03 2007

Status

approved