A362787 - OEIS

A362787

Triangle read by rows, T(n, k) = (-1)^k * RisingFactorial(n, k) * FallingFactorial(k - n, k).

1, 1, 0, 1, 2, 0, 1, 6, 24, 0, 1, 12, 120, 720, 0, 1, 20, 360, 5040, 40320, 0, 1, 30, 840, 20160, 362880, 3628800, 0, 1, 42, 1680, 60480, 1814400, 39916800, 479001600, 0, 1, 56, 3024, 151200, 6652800, 239500800, 6227020800, 87178291200, 0, 1, 72, 5040, 332640, 19958400, 1037836800, 43589145600, 1307674368000, 20922789888000, 0

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OFFSET

0,5

LINKS

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

FORMULA

T(n, k) = Pochhammer(n, k) * Pochhammer(n - k, k).

T(n, k) = Gamma(n + k) / Gamma(n - k) if k != n.

T(n, k) = (-1)^k * binomial(k - n, k) * binomial(n + k - 1, k) * (k!)^2.

T(n, k) = binomial(n + k - 1, n - 1) * binomial(n - 1, n - 1 - k) *(k!)^2.

EXAMPLE

Triangle T(n, k) starts:

[0] 1;

[1] 1, 0;

[2] 1, 2, 0;

[3] 1, 6, 24, 0;

[4] 1, 12, 120, 720, 0;

[5] 1, 20, 360, 5040, 40320, 0;

[6] 1, 30, 840, 20160, 362880, 3628800, 0;

[7] 1, 42, 1680, 60480, 1814400, 39916800, 479001600, 0;

[8] 1, 56, 3024, 151200, 6652800, 239500800, 6227020800, 87178291200, 0;

MAPLE

T := (n, k) -> if n = k then 0^k else GAMMA(n + k) / GAMMA(n - k) fi:

seq(seq(T(n, k), k = 0..n), n = 0..9);

MATHEMATICA

Table[Pochhammer[n, k]*Pochhammer[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 05 2023 *)

CROSSREFS

Cf. A362846 (row sums), A010050 (main diagonal), A002378 (column 1).

Sequence in context: A362588 A367073 A176129 * A341200 A300130 A101371

Adjacent sequences: A362784 A362785 A362786 * A362788 A362789 A362790

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 05 2023

STATUS

approved