OFFSET
0,4
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
CROSSREFS
KEYWORD
AUTHOR
F. Chapoton, Jan 27 2018
STATUS
approved