A298854 Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 1, 2, 3, 2, 6, 11, 11, 6, 24, 50, 61, 50, 24, 120, 274, 379, 379, 274, 120, 720, 1764, 2668, 3023, 2668, 1764, 720, 5040, 13068, 21160, 26193, 26193, 21160, 13068, 5040, 40320, 109584, 187388, 248092, 270961, 248092, 187388, 109584, 40320, 362880, 1026576, 1836396, 2565080, 2995125, 2995125, 2565080, 1836396, 1026576, 362880

Offset: 0

Author

F. Chapoton, Jan 27 2018

Keywords

• tabl
• nonn
• easy

Comments

This is just a different normalization of A223256 and A223257.

Examples

			For n = 3, the polynomial is 6*x^3 + 11*x^2 + 11*x + 6.
The first few polynomials, as a table:
[  1],
[  1,   1],
[  2,   3,   2],
[  6,  11,  11,   6],
[ 24,  50,  61,  50,  24],
[120, 274, 379, 379, 274, 120]
		

Crossrefs

Closely related to A223256 and A223257.

Row sums are A002720.

Leftmost and rightmost columns are A000142.

Alternating row sums are A177145.

Absolute value of evaluation at x = exp(2*i*Pi/3) is A080171.

Evaluation at x=2 gives A187735.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, n+1, expand(
      n*(x+1)*b(n-1)-(n-1)^2*x*b(n-2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 01 2021

Mathematica

P[0] = 1 ; P[1] = x + 1;
P[n_] := P[n] = n (x + 1) P[n - 1] - (n - 1)^2 x P[n - 2];
Table[CoefficientList[P[n], x], {n, 0, 9}] // Flatten (* Jean-François Alcover, Mar 16 2020 *)

Sage

@cached_function
def poly(n):
    x = polygen(ZZ, 'x')
    if n < 0:
        return x.parent().zero()
    elif n == 0:
        return x.parent().one()
    else:
        return n * (x + 1) * poly(n - 1) - (n - 1)**2 * x * poly(n - 2)
A298854_row = lambda n: list(poly(n))
for n in (0..7): print(A298854_row(n))

Formula

P(0)=1 and P(n) = n * (x + 1) * P(n - 1) - (n - 1)^2 * x * P(n - 2).