OFFSET
0,6
COMMENTS
The antidiagonal representation of the falling factorials (A008279).
FORMULA
T(n, k) = RisingFactorial(n + 1 - 2*k, k).
T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).
EXAMPLE
[ 0] [1]
[ 1] [1]
[ 2] [1, 1]
[ 3] [1, 2]
[ 4] [1, 3, 2]
[ 5] [1, 4, 6]
[ 6] [1, 5, 12, 6]
[ 7] [1, 6, 20, 24]
[ 8] [1, 7, 30, 60, 24]
[ 9] [1, 8, 42, 120, 120]
[10] [1, 9, 56, 210, 360, 120]
[11] [1, 10, 72, 336, 840, 720]
MAPLE
T := (n, k) -> pochhammer(n + 1 - 2*k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);
PROG
(Sage)
def T(n, k): return rising_factorial(n + 1 - 2*k, k)
def T(n, k): return (-1)^k*falling_factorial(2*k - n - 1, k)
def T(n, k): return binomial(n - k, k) * factorial(k)
print(flatten([[T(n, k) for k in (0..n//2)] for n in (0..11)]))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, May 17 2021
STATUS
approved