A132888 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 height k (0<=k<=floor(n/2)).

1, 1, 2, 1, 3, 3, 5, 10, 1, 8, 27, 5, 13, 72, 23, 1, 21, 183, 86, 7, 34, 459, 302, 40, 1, 55, 1134, 994, 185, 9, 89, 2782, 3158, 778, 61, 1, 144, 6786, 9742, 3027, 332, 11, 233, 16498, 29450, 11196, 1607, 86, 1, 377, 40014, 87602, 39795, 7117, 535, 13

Offset

0,3

Comments

Row n has 1+floor(n/2) terms. T(n,0)=A000045(n+1) (the Fibonacci numbers). Row sums yield A128720.

Links

Table of n, a(n) for n=0..55.

Formula

G.f. of column k is g[k]=G[k]-G[k-1], where G[0]:=1/(1-z-z^2), G[k]=1/(1-z-z^2-z^2*G[k-1]).

Example

T(3,1)=3 because we have hUD, UhD and UDh.
Triangle starts:
1;
1;
2,1;
3,3;
5,10,1;
8,27,5;
13,72,23,1;

Maple

G[0]:=1/(1-z-z^2): for n to 13 do G[n]:=simplify(1/(1-z-z^2-z^2*G[n-1])) end do: g[0]:=G[0]: for n to 13 do g[n]:=simplify(G[n]-G[n-1]) end do: for n from 0 to 13 do gser[n]:=series(g[n], z=0, 40) end do: for n from 0 to 13 do seq(coeff(gser[k], z, n), k=0..floor((1/2)*n)) end do;

Crossrefs

Cf. A000045, A128720.

Sequence in context: A059876 A095354 A132883 * A213934 A124774 A056610

Adjacent sequences: A132885 A132886 A132887 * A132889 A132890 A132891

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 05 2007

Status

approved