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

%I A110446 #7 Oct 08 2016 03:20:33
%S A110446 1,2,1,8,4,1,32,24,6,1,136,128,48,8,1,592,680,320,80,10,1,2624,3552,
%T A110446 2040,640,120,12,1,11776,18368,12432,4760,1120,168,14,1,53344,94208,
%U A110446 73472,33152,9520,1792,224,16,1,243392,480096,423936,220416,74592,17136
%N A110446 Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.
%C A110446 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 A110446 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 A110446 Table begins
%e A110446 \ k...0....1....2....3....4....
%e A110446 n\
%e A110446 0 |...1
%e A110446 1 |...2....1
%e A110446 2 |...8....4....1
%e A110446 3 |..32...24....6....1
%e A110446 4 |.136..128...48....8....1
%e A110446 5 |.592..680..320...80...10....1
%e A110446 The paths ENDD, NDDE, DEND, DNDE, DDEN, DDNE each have 2 Ds not preceded by an E,
%e A110446 and so T(3,2)=6.
%t A110446 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 A110446 Column k=0 is A006139.
%K A110446 nonn,tabl
%O A110446 0,2
%A A110446 _David Callan_, Jul 20 2005