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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). 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A036052 A308260 A279212 * A269830 A275312 A209026

Adjacent sequences:  A299015 A299016 A299017 * A299019 A299020 A299021

KEYWORD

tabl,nonn,easy,changed

AUTHOR

F. Chapoton, Jan 31 2018

STATUS

approved

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)