cp's OEIS Frontend

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

%I A128890 #10 May 22 2019 06:31:31
%S A128890 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,
%T A128890 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
%N A128890 Triangle T(n,k) related to walks in the positive quadrant.
%H A128890 G. C. Greubel, <a href="/A128890/b128890.txt">Rows n = 0..100 of triangle, flattened</a>
%F A128890 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).
%e A128890 Triangle begins:
%e A128890     1;
%e A128890     0,    1;
%e A128890     2,    0,   1;
%e A128890     0,    5,   0,    1;
%e A128890    10,    0,   9,    0,   1;
%e A128890     0,   35,   0,   14,   0,   1;
%e A128890    70,    0,  84,    0,  20,   0,  1;
%e A128890     0,  294,   0,  168,   0,  27,  0,  1;
%e A128890   588,    0, 840,    0, 300,   0, 35,  0,  1;
%e A128890     0, 2772,   0, 1980,   0, 495,  0, 44,  0,  1;
%t A128890 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]];
%t A128890 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
%o A128890 (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
%o A128890 (Sage)
%o A128890 def T(n, k):
%o A128890     if (k==0 and n%2==0): return 4*binomial(n,n/2)*binomial(n+2, (n+2)/2)/((n+2)*(n+4))
%o A128890     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)
%o A128890     else: return 0
%o A128890 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 20 2019
%Y A128890 Cf. A000012, A000096, A052472, A005568, A000356.
%K A128890 nonn,tabl
%O A128890 0,4
%A A128890 _Philippe Deléham_, Apr 20 2007