OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
EXAMPLE
Triangle begins as:
0;
-1, 1;
-1, -2, 1;
-4, 0, -3, 1;
-13, -4, 2, -4, 1;
-46, -10, -5, 5, -5, 1;
MATHEMATICA
T[n_, k_]:= If[n==k==0, 0, If[k==n, 1, n*Sum[(-1)^j*j*Binomial[j+k, k]* Binomial[2*n-2*k-j-1, n-k-1]/((j+k)*(n-k)), {j, 1, n-k}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2019 *)
PROG
(Maxima) T(n, k):=if n=0 and k=0 then 0 else if n=k then 1 else n*sum(binomial(i+k, k)*(i)*binomial(2*(n-k)-i-1, n-k-1)*(-1)^(i)/((i+k)*(n-k)), i, 1, n-k);
(PARI) {T(n, k) = if(n==0 && k==0, 0, if(k==n, 1, n*sum(j=1, n-k, (-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)))))}; \\ G. C. Greubel, Apr 29 2019
(Magma) [[n eq 0 and k eq 0 select 0 else k eq n select 1 else n*(&+[ (-1)^j*j*Binomial(j+k, k)*Binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)): j in [1..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 29 2019
(Sage)
def T(n, k):
if (k==n==0): return 0
elif (k==n): return 1
else: return n*sum((-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)) for j in (1..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 29 2019
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Dmitry Kruchinin, Jun 24 2013
STATUS
approved