A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 6, 1, 5, 12, 6, 1, 6, 20, 24, 1, 7, 30, 60, 24, 1, 8, 42, 120, 120, 1, 9, 56, 210, 360, 120, 1, 10, 72, 336, 840, 720, 1, 11, 90, 504, 1680, 2520, 720, 1, 12, 110, 720, 3024, 6720, 5040, 1, 13, 132, 990, 5040, 15120, 20160, 5040

Offset: 0

Author

Peter Luschny, May 17 2021

Keywords

• nonn
• tabf

Comments

The antidiagonal representation of the falling factorials (A008279).

Examples

			[ 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]
		

Crossrefs

Cf. A122852 (row sums).

Cf. A008279, A122851.

Programs

Maple

T := (n, k) -> pochhammer(n + 1 - 2*k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);

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)]))

Formula

T(n, k) = RisingFactorial(n + 1 - 2*k, k).

T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).