A128890 Triangle T(n,k) related to walks in the positive quadrant.

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 10, 0, 9, 0, 1, 0, 35, 0, 14, 0, 1, 70, 0, 84, 0, 20, 0, 1, 0, 294, 0, 168, 0, 27, 0, 1, 588, 0, 840, 0, 300, 0, 35, 0, 1, 0, 2772, 0, 1980, 0, 495, 0, 44, 0, 1

Offset: 0

Author

Philippe Deléham, Apr 20 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    0,    1;
    2,    0,   1;
    0,    5,   0,    1;
   10,    0,   9,    0,   1;
    0,   35,   0,   14,   0,   1;
   70,    0,  84,    0,  20,   0,  1;
    0,  294,   0,  168,   0,  27,  0,  1;
  588,    0, 840,    0, 300,   0, 35,  0,  1;
    0, 2772,   0, 1980,   0, 495,  0, 44,  0,  1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000012, A000096, A052472, A005568, A000356.

Programs

Mathematica

T[n_, k_]:= If[k==0 && EvenQ[n], 4*Binomial[n,n/2]*Binomial[n+2,(n+2)/2 ]/((n+2)*(n+4)), If[EvenQ[n+k], Binomial[n, (n+k)/2]*Binomial[n+2, (n - k)/2] - Binomial[n+2, (n+k+2)/2]*Binomial[n, (n-k-2)/2], 0]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

PARI

{ T(n,k) = if(k==0 && n%2==0, 4*binomial(n,n/2)*binomial(n+2, (n+2)/2)/((n+2)*(n+4)), if((n+k)%2==0, binomial(n, (n+k)/2)*binomial(n + 2, (n-k)/2) - binomial(n+2, (n+k+2)/2)*binomial(n, (n-k-2)/2), 0)) }; \\ G. C. Greubel, May 20 2019

Sage

def T(n, k):
    if (k==0 and n%2==0): return 4*binomial(n,n/2)*binomial(n+2, (n+2)/2)/((n+2)*(n+4))
    elif ((n+k)%2==0): return binomial(n, (n+k)/2)*binomial(n + 2, (n-k)/2) - binomial(n+2, (n+k+2)/2)*binomial(n, (n-k-2)/2)
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 20 2019

Formula

T(n,k) = binomial(n, r)*binomial(n+2, s) - binomial(n+2, r+1)*binomial(n, s-1) with r=(n+k)/2 and s=(n-k)/2, if n+k is even otherwise T(n,k)=0. Also T(2*n,0) = A000108(n)*A000108(n+1) = A005568(n).