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%I #34 Apr 01 2021 18:07:00
%S 1,1,1,2,3,2,6,11,11,6,24,50,61,50,24,120,274,379,379,274,120,720,
%T 1764,2668,3023,2668,1764,720,5040,13068,21160,26193,26193,21160,
%U 13068,5040,40320,109584,187388,248092,270961,248092,187388,109584,40320,362880,1026576,1836396,2565080,2995125,2995125,2565080,1836396,1026576,362880
%N Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.
%C This is just a different normalization of A223256 and A223257.
%F P(0)=1 and P(n) = n * (x + 1) * P(n - 1) - (n - 1)^2 * x * P(n - 2).
%e For n = 3, the polynomial is 6*x^3 + 11*x^2 + 11*x + 6.
%e The first few polynomials, as a table:
%e [ 1],
%e [ 1, 1],
%e [ 2, 3, 2],
%e [ 6, 11, 11, 6],
%e [ 24, 50, 61, 50, 24],
%e [120, 274, 379, 379, 274, 120]
%p b:= proc(n) option remember; `if`(n<1, n+1, expand(
%p n*(x+1)*b(n-1)-(n-1)^2*x*b(n-2)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Apr 01 2021
%t P[0] = 1 ; P[1] = x + 1;
%t P[n_] := P[n] = n (x + 1) P[n - 1] - (n - 1)^2 x P[n - 2];
%t Table[CoefficientList[P[n], x], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Mar 16 2020 *)
%o (Sage)
%o @cached_function
%o def poly(n):
%o x = polygen(ZZ, 'x')
%o if n < 0:
%o return x.parent().zero()
%o elif n == 0:
%o return x.parent().one()
%o else:
%o return n * (x + 1) * poly(n - 1) - (n - 1)**2 * x * poly(n - 2)
%o A298854_row = lambda n: list(poly(n))
%o for n in (0..7): print(A298854_row(n))
%Y Closely related to A223256 and A223257.
%Y Row sums are A002720.
%Y Leftmost and rightmost columns are A000142.
%Y Alternating row sums are A177145.
%Y Absolute value of evaluation at x = exp(2*i*Pi/3) is A080171.
%Y Evaluation at x=2 gives A187735.
%K tabl,nonn,easy
%O 0,4
%A _F. Chapoton_, Jan 27 2018
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