OFFSET
0,5
COMMENTS
A090443 is defined as +1 at negative indices here, which keeps the definition valid in the range 0 <= k <= n.
Row sums are 1, 2, 8, 50, 362, 2642, 19322, 141794, 1045298, 7742882, ....
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = 2*binomial(n-1,k-1)*binomial(n,k)*binomial(n+1,k+1)*(n-k)/(n-k+1) with T(n, 0) = T(n, n) = 1.
T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} (j-1)*j*(j+1) = (n-1)!*n!*(n+1)!/2 and c(0) = c(1) = 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 24, 24, 1;
1, 60, 240, 60, 1;
1, 120, 1200, 1200, 120, 1;
1, 210, 4200, 10500, 4200, 210, 1;
1, 336, 11760, 58800, 58800, 11760, 336, 1;
1, 504, 28224, 246960, 493920, 246960, 28224, 504, 1;
1, 720, 60480, 846720, 2963520, 2963520, 846720, 60480, 720, 1;
1, 990, 118800, 2494800, 13970880, 24449040, 13970880, 2494800, 118800, 990, 1;
...
MAPLE
MATHEMATICA
T[n_, k_]:= If[k==0||k==n, 1, 2*Binomial[n-1, k-1]*Binomial[n, k]*Binomial[n+1, k+1]*(n-k)/(n-k+1)];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 17 2021 *)
PROG
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else 2*Binomial(n-1, k-1)*Binomial(n, k)*Binomial(n+1, k+1)*(n-k)/(n-k+1) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 17 2021
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n-1, k-1)*binomial(n, k)*binomial(n+1, k+1)*(n-k)/(n-k+1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 17 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 01 2010
STATUS
approved