proposed

approved

editing

proposed

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 *)

approved

editing

_David Callan (callan(AT)stat.wisc.edu), _, Jul 20 2005

GfG.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)

nonn,tabl,new

Gf. 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)

nonn,tabl,new

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

0,2

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.

Gf. 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)

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.

Column k=0 is A006139.

nonn,tabl

David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

approved