A349971 Array read by ascending antidiagonals, A(n, k) = -(-n)^k*FallingFactorial(1/n, k) for n, k >= 1.
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, 1, 6, 55, 504, 3465, 12320, 10395, 0, 1, 7, 78, 935, 9576, 65835, 209440, 135135, 0, 1, 8, 105, 1560, 21505, 229824, 1514205, 4188800, 2027025, 0
Offset: 1
Examples
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]
Links
- G. C. Greubel, Antidiagonals n = 1..50, flattened
Crossrefs
Programs
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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
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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 *) -
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)])
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)