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A349971 Array read by ascending antidiagonals, A(n, k) = -(-n)^k*FallingFactorial(1/n, k) for n, k >= 1.

%I A349971 #17 Mar 01 2022 01:26:18
%S A349971 1,1,0,1,1,0,1,2,3,0,1,3,10,15,0,1,4,21,80,105,0,1,5,36,231,880,945,0,
%T A349971 1,6,55,504,3465,12320,10395,0,1,7,78,935,9576,65835,209440,135135,0,
%U A349971 1,8,105,1560,21505,229824,1514205,4188800,2027025,0
%N A349971 Array read by ascending antidiagonals, A(n, k) = -(-n)^k*FallingFactorial(1/n, k) for n, k >= 1.
%H A349971 G. C. Greubel, <a href="/A349971/b349971.txt">Antidiagonals n = 1..50, flattened</a>
%F A349971 From _G. C. Greubel_, Feb 22 2022: (Start)
%F A349971 A(n, k) = n^(k-1)*Pochhammer((n-1)/n, k-1) (array).
%F A349971 T(n, k) = (n-k+1)^(k-1)*Pochhammer((n-k)/(n-k+1), k-1) (antidiagonal triangle).
%F A349971 T(2*n, n) = (-1)^(n-1)*A158886(n). (End)
%e A349971 Array starts:
%e A349971 [1] 1, 0,   0,    0,      0,       0,         0,           0, ... A000007
%e A349971 [2] 1, 1,   3,   15,    105,     945,     10395,      135135, ... A001147
%e A349971 [3] 1, 2,  10,   80,    880,   12320,    209440,     4188800, ... A008544
%e A349971 [4] 1, 3,  21,  231,   3465,   65835,   1514205,    40883535, ... A008545
%e A349971 [5] 1, 4,  36,  504,   9576,  229824,   6664896,   226606464, ... A008546
%e A349971 [6] 1, 5,  55,  935,  21505,  623645,  21827575,   894930575, ... A008543
%e A349971 [7] 1, 6,  78, 1560,  42120, 1432080,  58715280,  2818333440, ... A049209
%e A349971 [8] 1, 7, 105, 2415,  74865, 2919735, 137227545,  7547514975, ... A049210
%e A349971 [9] 1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, ... A049211
%e A349971 Triangle starts:
%e A349971 [1] [1]
%e A349971 [2] [1, 0]
%e A349971 [3] [1, 1,  0]
%e A349971 [4] [1, 2,  3,   0]
%e A349971 [5] [1, 3, 10,  15,    0]
%e A349971 [6] [1, 4, 21,  80,  105,     0]
%e A349971 [7] [1, 5, 36, 231,  880,   945,      0]
%e A349971 [8] [1, 6, 55, 504, 3465, 12320,  10395,      0]
%e A349971 [9] [1, 7, 78, 935, 9576, 65835, 209440, 135135, 0]
%t A349971 A[n_, k_] := -(-n)^k * FactorialPower[1/n, k]; Table[A[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Dec 21 2021 *)
%o A349971 (SageMath)
%o A349971 def A(n, k): return -(-n)^k*falling_factorial(1/n, k)
%o A349971 def T(n, k): return A(n-k+1, k)
%o A349971 for n in (1..9): print([A(n, k) for k in (1..8)])
%o A349971 for n in (1..9): print([T(n, k) for k in (1..n)])
%o A349971 (Magma) [k eq n select 0^(n-1) else Round((n-k+1)^(k-1)*Gamma(k-1 + (n-k)/(n-k+1))/Gamma((n-k)/(n-k+1))): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Feb 22 2022
%Y A349971 Rows 1-9: A000007, A001147, A008544, A008545, A008546, A008543, A049209, A049210, A049211.
%Y A349971 Main diagonal A349731.
%Y A349971 Cf. A158886, A256268.
%K A349971 nonn,tabl
%O A349971 1,8
%A A349971 _Peter Luschny_, Dec 21 2021