A099599 Triangle T read by rows: coefficients of polynomials generating array A099597.

1, 1, 1, 1, 0, 2, 1, 9, -12, 6, 1, -104, 204, -120, 24, 1, 2265, -4840, 3540, -1080, 120, 1, -71064, 164910, -138840, 54360, -10080, 720, 1, 3079825, -7626948, 7134330, -3300360, 808920, -100800, 5040, 1, -176449776, 460982648, -468313104, 244938960, -72266880, 12156480, -1088640, 40320

Offset

0,6

Comments

Row sums are k (A000027), left edge columns are factorials (A000142). [Peter Bala, Aug 19 2013]

Links

Table of n, a(n) for n=0..44.

Formula

The row polynomials satisfy the second order recurrence equation R(n,x) = (n*x+1)*R(n-1,x-1) - (n-1)*(x-1)*R(n-2,x-2), with the initial conditions R(0,x) = 1 and R(1,x) = 1+x. - Peter Bala, Aug 19 2013

Example

1
x + 1
2*x^2 + 1
6*x^3 - 12*x^2 + 9*x + 1
24*x^4 - 120*x^3 + 204*x^2 - 104*x + 1
120*x^5 - 1080*x^4 + 3540*x^3 - 4840*x^2 + 2265*x + 1

Maple

# Define row polynomials R(n, x) recursively:
R := proc(n, x) option remember; if n = 0 then 1 elif n = 1 then 1+x
else (n*x+1)*procname(n-1, x-1) - (n-1)*(x-1)*procname(n-2, x-2) fi end:
Trow := n -> PolynomialTools:-CoefficientList(R(n, x), x);
seq(Trow(n), n = 0..10); # Peter Bala, Aug 19 2013

Mathematica

R[n_, x_] := R[n, x] = (n x + 1) R[n-1, x-1] - (n-1) (x-1) R[n-2, x-2]; R[0, _] = 1; R[1, x_] = 1 + x;
Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)

Crossrefs

Sequence in context: A240085 A078623 A198204 * A085488 A072265 A192352

Adjacent sequences: A099596 A099597 A099598 * A099600 A099601 A099602

Keyword

sign,tabl

Author

Ralf Stephan, Oct 28 2004

Status

approved