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

0,6

Comments

The antidiagonal representation of the falling factorials (A008279).

Links

Table of n, a(n) for n=0..63.

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

Cf. A122852 (row sums).

Cf. A008279, A122851.

Sequence in context: A277813 A200154 A208825 * A089353 A136451 A361894

Adjacent sequences: A344388 A344389 A344390 * A344392 A344393 A344394

Keyword

nonn,tabf

Author

Peter Luschny, May 17 2021

Status

approved