A223256
Triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the numerator of the coefficient of x^k in the characteristic polynomial of the matrix realizing the transformation to Jacobi coordinates for a system of n particles on a line.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 25, 61, 25, 1, 1, 137, 379, 379, 137, 1, 1, 49, 667, 3023, 667, 49, 1, 1, 363, 529, 8731, 8731, 529, 363, 1, 1, 761, 46847, 62023, 270961, 62023, 46847, 761, 1, 1, 7129, 51011, 9161, 28525, 28525, 9161, 51011, 7129, 1
Offset: 0
Triangle begins:
1,
1, 1,
1, 3, 1,
1, 11, 11, 1,
1, 25, 61, 25, 1,
1, 137, 379, 379, 137, 1,
1, 49, 667, 3023, 667, 49, 1,
1, 363, 529, 8731, 8731, 529, 363, 1,
...
A298854
Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
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
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]
Leftmost and rightmost columns are
A000142.
Absolute value of evaluation at x = exp(2*i*Pi/3) is
A080171.
-
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
-
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 *)
-
@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))
Showing 1-2 of 2 results.
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