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

Author

Peter Luschny, Dec 21 2021

Keywords

• nonn
• tabl

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

Rows 1-9: A000007, A001147, A008544, A008545, A008546, A008543, A049209, A049210, A049211.

Main diagonal A349731.

Cf. A158886, A256268.

Programs

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

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)