A132884 Triangle read by rows: T(n,k) is the number of paths in the right half-plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k h=(1,0) steps (0<=k<=n).

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 13, 0, 15, 0, 1, 0, 57, 0, 24, 0, 1, 63, 0, 156, 0, 35, 0, 1, 0, 384, 0, 340, 0, 48, 0, 1, 321, 0, 1380, 0, 645, 0, 63, 0, 1, 0, 2505, 0, 3800, 0, 1113, 0, 80, 0, 1, 1683, 0, 11145, 0, 8855, 0, 1792, 0, 99, 0, 1, 0, 16008, 0, 37065, 0, 18368, 0, 2736, 0

Offset: 0

Author

Emeric Deutsch, Sep 03 2007

Keywords

• nonn
• tabl

Comments

T(2n,0)=A001850(n) (the central Delannoy numbers); T(2n+1,0)=0. T(2n,1)=0; T(2n-1,1)=A108666(n). T(n,k)=0 if n+k is odd. Row sums yield A059345. See A132277 for the same statistic on paths restricted to the first quadrant.

Examples

			Triangle starts:
   1;
   0,  1;
   3,  0,  1;
   0,  8,  0,  1;
  13,  0, 15,  0,  1;
   0, 57,  0, 24,  0,  1;
T(3,1)=8 because we have hH, Hh, hUD, UhD, UDh, hDU, DhU and DUh.
		

Crossrefs

Cf. A001850, A108666, A059345, A132277.

Programs

Maple

G:=1/sqrt((1-t*z-z^2)^2-4*z^2): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form

Formula

G.f. = 1/sqrt((1-tz-z^2)^2-4z^2).