A349728 - OEIS

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

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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