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

0,4

Links

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

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).

Example

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;

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

Prog

(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

Crossrefs

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

Sequence in context: A132277 A137286 A180048 * A196777 A318361 A078924

Adjacent sequences: A128887 A128888 A128889 * A128891 A128892 A128893

Keyword

nonn,tabl

Author

Philippe Deléham, Apr 20 2007

Status

approved