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A298854 Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.

%I A298854 #34 Apr 01 2021 18:07:00
%S A298854 1,1,1,2,3,2,6,11,11,6,24,50,61,50,24,120,274,379,379,274,120,720,
%T A298854 1764,2668,3023,2668,1764,720,5040,13068,21160,26193,26193,21160,
%U A298854 13068,5040,40320,109584,187388,248092,270961,248092,187388,109584,40320,362880,1026576,1836396,2565080,2995125,2995125,2565080,1836396,1026576,362880
%N A298854 Characteristic polynomials of Jacobi coordinates. Triangle read by rows, T(n, k) for 0 <= k <= n.
%C A298854 This is just a different normalization of A223256 and A223257.
%F A298854 P(0)=1 and P(n) = n * (x + 1) * P(n - 1) - (n - 1)^2 * x * P(n - 2).
%e A298854 For n = 3, the polynomial is 6*x^3 + 11*x^2 + 11*x + 6.
%e A298854 The first few polynomials, as a table:
%e A298854 [  1],
%e A298854 [  1,   1],
%e A298854 [  2,   3,   2],
%e A298854 [  6,  11,  11,   6],
%e A298854 [ 24,  50,  61,  50,  24],
%e A298854 [120, 274, 379, 379, 274, 120]
%p A298854 b:= proc(n) option remember; `if`(n<1, n+1, expand(
%p A298854       n*(x+1)*b(n-1)-(n-1)^2*x*b(n-2)))
%p A298854     end:
%p A298854 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p A298854 seq(T(n), n=0..10);  # _Alois P. Heinz_, Apr 01 2021
%t A298854 P[0] = 1 ; P[1] = x + 1;
%t A298854 P[n_] := P[n] = n (x + 1) P[n - 1] - (n - 1)^2 x P[n - 2];
%t A298854 Table[CoefficientList[P[n], x], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Mar 16 2020 *)
%o A298854 (Sage)
%o A298854 @cached_function
%o A298854 def poly(n):
%o A298854     x = polygen(ZZ, 'x')
%o A298854     if n < 0:
%o A298854         return x.parent().zero()
%o A298854     elif n == 0:
%o A298854         return x.parent().one()
%o A298854     else:
%o A298854         return n * (x + 1) * poly(n - 1) - (n - 1)**2 * x * poly(n - 2)
%o A298854 A298854_row = lambda n: list(poly(n))
%o A298854 for n in (0..7): print(A298854_row(n))
%Y A298854 Closely related to A223256 and A223257.
%Y A298854 Row sums are A002720.
%Y A298854 Leftmost and rightmost columns are A000142.
%Y A298854 Alternating row sums are A177145.
%Y A298854 Absolute value of evaluation at x = exp(2*i*Pi/3) is A080171.
%Y A298854 Evaluation at x=2 gives A187735.
%K A298854 tabl,nonn,easy
%O A298854 0,4
%A A298854 _F. Chapoton_, Jan 27 2018