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%I #14 May 18 2021 07:26:09
%S 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,
%T 120,1,9,56,210,360,120,1,10,72,336,840,720,1,11,90,504,1680,2520,720,
%U 1,12,110,720,3024,6720,5040,1,13,132,990,5040,15120,20160,5040
%N T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.
%C The antidiagonal representation of the falling factorials (A008279).
%F T(n, k) = RisingFactorial(n + 1 - 2*k, k).
%F T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).
%e [ 0] [1]
%e [ 1] [1]
%e [ 2] [1, 1]
%e [ 3] [1, 2]
%e [ 4] [1, 3, 2]
%e [ 5] [1, 4, 6]
%e [ 6] [1, 5, 12, 6]
%e [ 7] [1, 6, 20, 24]
%e [ 8] [1, 7, 30, 60, 24]
%e [ 9] [1, 8, 42, 120, 120]
%e [10] [1, 9, 56, 210, 360, 120]
%e [11] [1, 10, 72, 336, 840, 720]
%p T := (n, k) -> pochhammer(n + 1 - 2*k, k):
%p seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);
%o (Sage)
%o def T(n, k): return rising_factorial(n + 1 - 2*k, k)
%o def T(n, k): return (-1)^k*falling_factorial(2*k - n - 1, k)
%o def T(n, k): return binomial(n - k, k) * factorial(k)
%o print(flatten([[T(n, k) for k in (0..n//2)] for n in (0..11)]))
%Y Cf. A122852 (row sums).
%Y Cf. A008279, A122851.
%K nonn,tabf
%O 0,6
%A _Peter Luschny_, May 17 2021
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