cp's OEIS Frontend

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

%I A344391 #14 May 18 2021 07:26:09
%S A344391 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 A344391 120,1,9,56,210,360,120,1,10,72,336,840,720,1,11,90,504,1680,2520,720,
%U A344391 1,12,110,720,3024,6720,5040,1,13,132,990,5040,15120,20160,5040
%N A344391 T(n, k) = binomial(n - k, k) * factorial(k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.
%C A344391 The antidiagonal representation of the falling factorials (A008279).
%F A344391 T(n, k) = RisingFactorial(n + 1 - 2*k, k).
%F A344391 T(n, k) = (-1)^k*FallingFactorial(2*k - n - 1, k).
%e A344391 [ 0] [1]
%e A344391 [ 1] [1]
%e A344391 [ 2] [1,  1]
%e A344391 [ 3] [1,  2]
%e A344391 [ 4] [1,  3,  2]
%e A344391 [ 5] [1,  4,  6]
%e A344391 [ 6] [1,  5, 12,   6]
%e A344391 [ 7] [1,  6, 20,  24]
%e A344391 [ 8] [1,  7, 30,  60,  24]
%e A344391 [ 9] [1,  8, 42, 120, 120]
%e A344391 [10] [1,  9, 56, 210, 360, 120]
%e A344391 [11] [1, 10, 72, 336, 840, 720]
%p A344391 T := (n, k) -> pochhammer(n + 1 - 2*k, k):
%p A344391 seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);
%o A344391 (Sage)
%o A344391 def T(n, k): return rising_factorial(n + 1 - 2*k, k)
%o A344391 def T(n, k): return (-1)^k*falling_factorial(2*k - n - 1, k)
%o A344391 def T(n, k): return binomial(n - k, k) * factorial(k)
%o A344391 print(flatten([[T(n, k) for k in (0..n//2)] for n in (0..11)]))
%Y A344391 Cf. A122852 (row sums).
%Y A344391 Cf. A008279, A122851.
%K A344391 nonn,tabf
%O A344391 0,6
%A A344391 _Peter Luschny_, May 17 2021