A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n*(x + 1)*P(n - 1) - (n - 2)^2*x*P(n - 2).

1, 2, 2, 6, 11, 6, 24, 60, 60, 24, 120, 366, 501, 366, 120, 720, 2532, 4242, 4242, 2532, 720, 5040, 19764, 38268, 46863, 38268, 19764, 5040, 40320, 172512, 373104, 528336, 528336, 373104, 172512, 40320, 362880, 1668528, 3942108, 6237828, 7213761, 6237828, 3942108, 1668528, 362880

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Formula

P(0) = 0, P(1) = 1 and P(n) = n * (x + 1) * P(n - 1) - (n - 2)^2 * x * P(n - 2).

Example

For n = 3, the polynomial is 6*x^2 + 11*x + 6.
The first few polynomials, as a table:
[1],
[2,    2],
[6,    11,    6],
[24,   60,    60,  24],
[120,  366,   501, 366, 120]

Maple

P:= proc(n) option remember; expand(`if`(n<2, n,
      n*(x+1)*P(n-1)-(n-2)^2*x*P(n-2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(P(n)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jan 31 2018
A := proc(n, k) ## n >= 0 and k = 0 .. n
    option remember;
    if n = 0 and k = 0 then
        1
    elif n > 0 and k >= 0 and k <= n then
        (n+1)*(A(n-1, k)+A(n-1, k-1))-(n-1)^2*A(n-2, k-1)
    else
        0
    end if;
end proc: # Yu-Sheng Chang, Apr 14 2020

Mathematica

P[n_] := P[n] = Expand[If[n < 2, n, n (x+1) P[n-1] - (n-2)^2 x P[n-2]]];
row[n_] := CoefficientList[P[n], x];
row /@ Range[12] // Flatten (* Jean-François Alcover, Dec 10 2019 *)

Prog

(Sage)
@cached_function
def poly(n):
    x = polygen(ZZ, 'x')
    if n < 1:
        return x.parent().zero()
    elif n == 1:
        return x.parent().one()
    else:
        return n * (x + 1) * poly(n - 1) - (n - 2)**2 * x * poly(n - 2)

Crossrefs

Very similar to A298854.

Row sums are A277382(n-1) for n>0.

Leftmost and rightmost columns are A000142.

Alternating row sums are A177145.

Alternating row sum of row 2*n+1 is A001818(n).

Sequence in context: A355207 A308260 A279212 * A364834 A269830 A350114

Adjacent sequences: A299015 A299016 A299017 * A299019 A299020 A299021

Keyword

tabl,nonn,easy

Author

F. Chapoton, Jan 31 2018

Status

approved