A223257 -id:A223257

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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.

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

The matrix J(n) realizing the change of coordinates for n particles is

[1, -1, 0, 0, 0, ... 0],

[1/2, 1/2, -1, 0, ... 0],

[1/3, 1/3, 1/3, -1, 0 ... 0],

...

[1/n, 1/n, 1/n, 1/n, ... 1/n]

Diagonals T(n,1)=T(n,n-1) are A001008, corresponding to the fact that the matrix J(n) above has trace equal to the n-th harmonic number.

See A223257 for denominators.

LINKS

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

Wikipedia, Jacobi coordinates

EXAMPLE

Triangle begins:

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,

...

KEYWORD

easy,frac,nonn,tabl

AUTHOR

Alberto Tacchella, Mar 18 2013

STATUS

approved

A298854

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

+10
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]

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.

KEYWORD

tabl,nonn,easy

AUTHOR

F. Chapoton, Jan 27 2018

STATUS

approved

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