A349728 Triangle read by rows, T(n, k) = RisingFactorial(k, n) / FallingFactorial(n, k).

1, 0, 1, 0, 1, 3, 0, 2, 4, 10, 0, 6, 10, 15, 35, 0, 24, 36, 42, 56, 126, 0, 120, 168, 168, 168, 210, 462, 0, 720, 960, 864, 720, 660, 792, 1716, 0, 5040, 6480, 5400, 3960, 2970, 2574, 3003, 6435, 0, 40320, 50400, 39600, 26400, 17160, 12012, 10010, 11440, 24310

Offset

0,6

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = (Gamma(1 - k + n)*Gamma(k + n))/(Gamma(k)*Gamma(1 + n)) for n >= 1.

Example

[0] [1]
[1] [0, 1]
[2] [0, 1,     3]
[3] [0, 2,     4,     10]
[4] [0, 6,     10,    15,    35]
[5] [0, 24,    36,    42,    56,    126]
[6] [0, 120,   168,   168,   168,   210,   462]
[7] [0, 720,   960,   864,   720,   660,   792,   1716]
[8] [0, 5040,  6480,  5400,  3960,  2970,  2574,  3003,  6435]
[9] [0, 40320, 50400, 39600, 26400, 17160, 12012, 10010, 11440, 24310]

Mathematica

T[n_, k_] := If[n == 0, 1, (Gamma[1 - k + n] Gamma[k + n])/(Gamma[k] Gamma[1 + n])]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Prog

(SageMath)
def T(n, k): return rising_factorial(k, n) // falling_factorial(n, k)
for n in range(10): print([T(n, k) for k in range(n + 1)])

Crossrefs

T(n, n) = A088218(n), row sums are A349730.

Sequence in context: A177344 A139585 A273084 * A261163 A292244 A208329

Adjacent sequences: A349725 A349726 A349727 * A349729 A349730 A349731

Keyword

nonn,tabl

Author

Peter Luschny, Nov 27 2021

Status

approved