OFFSET
0,8
LINKS
FORMULA
T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 0, 2, 6, 12, 20, ...
0, -1, 0, 9, 32, 75, ...
0, -1, -4, 9, 80, 275, ...
0, 0, -8, 0, 192, 1000, ...
MAPLE
T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021
MATHEMATICA
T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
PROG
(PARI) T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));
(PARI) T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));
(PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Feb 28 2021
STATUS
approved