2,3

Column T(n, 1) was omitted in the presentation of this sequence. Its definition may depend on the usage. In the combinatorics of subsequences it may be convenient to define T(n, 1) = 0, but if this sequence will be interpreted differently for example as polynomial coefficients then T(n, 1) = -n*n! could be a mathematically more natural definition.

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

An alternative definition of T(n, k) which includes k < 2 can be done by Laguerre polynomials:

Sum_{k=0..n} T(n, k)*x^k = n!*((1 + x)^n - 2*L_{n}(-x)), where L_{n} is the n-th Laguerre polynomial.

The triangle begins:

n| k: 2| 3| 4| 5| 6| 7|

==============================================

[2] 0,

[3] 0, 4

[4] 0, 64, 22

[5] 0, 800, 550, 118

[6] 0, 9600, 9900, 4248, 718

[7] 0, 117600, 161700, 104076, 35182, 5038

.

T(3, 3) = 4 because:

{1, 2, 3} has no subsequences which are neither increasing or decreasing.

{1, 3, 2} has {1, 3, 2}

{2, 1, 3} has {2, 1, 3}

{2, 3, 1} has {2, 3, 1}

{3, 1, 2} has {3, 1, 2}

{3, 2, 1} has no subsequences which are neither increasing or decreasing.

(PARI) T(n, k) = n!*binomial(n, k)-2*((n-k)! * binomial(n, n-k)^2)

(PARI) row(n) = if(n==2, [0], abs(Vecrev(-n!*((-1)^n*2*pollaguerre(n)-(-1+x)^n))[3..n+1]))

Cf. A144084.

Sequence in context: A247119 A296439 A228557 * A358294 A298616 A366828

Adjacent sequences: A373763 A373764 A373765 * A373767 A373768 A373769

nonn,tabl

Thomas Scheuerle, Jun 18 2024

approved