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A298854 Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This is just a different normalization of A223256 and A223257.

LINKS

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

FORMULA

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

EXAMPLE

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]

PROG

(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))

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.

Sequence in context: A110777 A087454 A059446 * A188881 A143806 A276551

Adjacent sequences:  A298851 A298852 A298853 * A298855 A298856 A298857

KEYWORD

tabl,nonn,easy,changed

AUTHOR

F. Chapoton, Jan 27 2018

STATUS

approved

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Last modified October 16 05:35 EDT 2019. Contains 328044 sequences. (Running on oeis4.)