OFFSET
1,8
LINKS
G. C. Greubel, Antidiagonals n = 1..50, flattened
FORMULA
From G. C. Greubel, Feb 22 2022: (Start)
A(n, k) = n^(k-1)*Pochhammer((n-1)/n, k-1) (array).
T(n, k) = (n-k+1)^(k-1)*Pochhammer((n-k)/(n-k+1), k-1) (antidiagonal triangle).
T(2*n, n) = (-1)^(n-1)*A158886(n). (End)
EXAMPLE
Array starts:
[1] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007
[2] 1, 1, 3, 15, 105, 945, 10395, 135135, ... A001147
[3] 1, 2, 10, 80, 880, 12320, 209440, 4188800, ... A008544
[4] 1, 3, 21, 231, 3465, 65835, 1514205, 40883535, ... A008545
[5] 1, 4, 36, 504, 9576, 229824, 6664896, 226606464, ... A008546
[6] 1, 5, 55, 935, 21505, 623645, 21827575, 894930575, ... A008543
[7] 1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, ... A049209
[8] 1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, ... A049210
[9] 1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, ... A049211
Triangle starts:
[1] [1]
[2] [1, 0]
[3] [1, 1, 0]
[4] [1, 2, 3, 0]
[5] [1, 3, 10, 15, 0]
[6] [1, 4, 21, 80, 105, 0]
[7] [1, 5, 36, 231, 880, 945, 0]
[8] [1, 6, 55, 504, 3465, 12320, 10395, 0]
[9] [1, 7, 78, 935, 9576, 65835, 209440, 135135, 0]
MATHEMATICA
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 *)
PROG
(SageMath)
def A(n, k): return -(-n)^k*falling_factorial(1/n, k)
def T(n, k): return A(n-k+1, k)
for n in (1..9): print([A(n, k) for k in (1..8)])
for n in (1..9): print([T(n, k) for k in (1..n)])
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 21 2021
STATUS
approved