A110446 Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.

1, 2, 1, 8, 4, 1, 32, 24, 6, 1, 136, 128, 48, 8, 1, 592, 680, 320, 80, 10, 1, 2624, 3552, 2040, 640, 120, 12, 1, 11776, 18368, 12432, 4760, 1120, 168, 14, 1, 53344, 94208, 73472, 33152, 9520, 1792, 224, 16, 1, 243392, 480096, 423936, 220416, 74592, 17136

Offset

0,2

Comments

T(n,k) = number of Delannoy paths (A001850) of steps east(E), north(N) and diagonal (D) (i.e., northeast) from (0,0) to (n,n) containing k Ds not preceded by an E.

Links

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

Formula

G.f. G(z, t)=Sum_{n>=k>=0}T(n, k)*z^n*t^k is given by G(z, t)= (1 - z(4 + 2*t) - z^2(4 - 4*t - t^2))^(-1/2)

Example

Table begins
\ k...0....1....2....3....4....
n\
0 |...1
1 |...2....1
2 |...8....4....1
3 |..32...24....6....1
4 |.136..128...48....8....1
5 |.592..680..320...80...10....1
The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,
and so T(3,2)=6.

Mathematica

T[n_, k_] := SeriesCoefficient[(1-z(4 + 2*t) - z^2 (4 - 4*t - t^2))^(-1/2), {z, 0, n}, {t, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 08 2016 *)

Crossrefs

Column k=0 is A006139.

Sequence in context: A110107 A154537 A201641 * A109979 A110171 A104988

Adjacent sequences: A110443 A110444 A110445 * A110447 A110448 A110449

Keyword

nonn,tabl

Author

David Callan, Jul 20 2005

Status

approved