A132280 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 H steps (0<=k<=floor(n/2)).

1, 1, 2, 1, 4, 2, 9, 6, 1, 21, 16, 3, 51, 45, 12, 1, 127, 126, 40, 4, 323, 357, 135, 20, 1, 835, 1016, 441, 80, 5, 2188, 2907, 1428, 315, 30, 1, 5798, 8350, 4572, 1176, 140, 6, 15511, 24068, 14535, 4284, 630, 42, 1, 41835, 69576, 45925, 15240, 2646, 224, 7

Offset: 0

Author

Emeric Deutsch, Sep 03 2007

Keywords

• nonn
• tabf

Comments

Also, 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 peaks (i.e., UD's). Row n contains 1+floor(n/2) terms. T(n,0) = A001006(n) (the Motzkin numbers). Row sums yield A128720. Sum(k*T(n,k), k>=0) = A106053(n).

Shifting every column k of this triangle k steps upwards yields triangle A091869, but with another offset. - Werner Schulte, Feb 03 2017

Examples

			Triangle starts:
1;
1;
2,1;
4,2;
9,6,1;
21,16,3;
51,45,12,1;
T(5,2)=3 because we have hHH, HhH and HHh.
		

Crossrefs

Cf. A001006, A106053, A128720.

Programs

Maple

G:=((1-z-t*z^2-sqrt((1+z-t*z^2)*(1-3*z-t*z^2)))*1/2)/z^2: Gser:=simplify(series(G,z=0,17)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n),n=0..13) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f. = G = G(t,z) satisfies G = 1+zG+tz^2*G+z^2*G^2 (see explicit expression at the Maple program).

Extensions

Keyword tabf added by Michel Marcus, Apr 09 2013