A110446 - OEIS

%I #7 Oct 08 2016 03:20:33

%S 1,2,1,8,4,1,32,24,6,1,136,128,48,8,1,592,680,320,80,10,1,2624,3552,

%T 2040,640,120,12,1,11776,18368,12432,4760,1120,168,14,1,53344,94208,

%U 73472,33152,9520,1792,224,16,1,243392,480096,423936,220416,74592,17136

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

%C 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.

%F 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)

%e Table begins

%e \ k...0....1....2....3....4....

%e n\

%e 0 |...1

%e 1 |...2....1

%e 2 |...8....4....1

%e 3 |..32...24....6....1

%e 4 |.136..128...48....8....1

%e 5 |.592..680..320...80...10....1

%e The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,

%e and so T(3,2)=6.

%t 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 *)

%Y Column k=0 is A006139.

%K nonn,tabl

%O 0,2

%A _David Callan_, Jul 20 2005