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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
OFFSET
0,4
COMMENTS
This is just a different normalization of A223256 and A223257.
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]
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 *)
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.
Evaluation at x=2 gives A187735.
Sequence in context: A110777 A087454 A059446 * A355266 A188881 A143806
KEYWORD
tabl,nonn,easy
AUTHOR
F. Chapoton, Jan 27 2018
STATUS
approved