A363826 Triangular array, read by rows: T(n,k) = coefficients of the polynomial (-1)^(n+1)/(n+1)! N(x), where N(x) is the numerator of the (n-1)st derivative of 1/(1-x-x^2), for k = 1..n.

1, 1, 2, 2, 3, 3, 3, 8, 6, 4, 1, 3, 4, 2, 1, 8, 30, 45, 40, 15, 6, 13, 56, 105, 105, 70, 21, 7, 21, 104, 224, 280, 210, 112, 28, 8, 34, 189, 468, 672, 630, 378, 168, 36, 9, 11, 68, 189, 312, 336, 252, 126, 48, 9, 2, 89, 605, 1870, 3465, 4290, 3696, 2310, 990

Offset: 1

Author

Clark Kimberling, Nov 26 2023

Keywords

• nonn
• tabl

Comments

The polynomials N(x) form a strong divisibility sequence. Multiplying every 5th polynomial by 5 results in another strong divisibility sequence of polynomials, F(n,x), in a Comment in A094440.

Examples

			First eleven rows:
   1
   1    2
   2    3    3
   3    8    6    4
   1    3    4    2    1
   8   30   45   40   15    6
  13   56  105  105   70   21    7
  21  104  224  280  210  112   28   8
  34  189  468  672  630  378  168  36   9
  11   68  189  312  336  252  126  48   9  2
  89  605 1870 3465 4290 3696 2310 990 330 55 11
Row 3 represents the polynomial 2 + 3*x + 3*x^2, extracted from
f"(x) = -((2*(2 + 3*x + 3*x^2))/(-1 + x + x^2)^3), where f(x) = 1/(1-x-x^2).
		

Crossrefs

Cf. A094440.

Programs

Mathematica

t = Table[CoefficientList[((-1)^(n + 1)) Numerator[Factor[D[1/(1 - x - x^2), {x, n}]]/(n + 1)!], x], {n, 0, 10}]
TableForm[t] (* array *)
Flatten[t ]  (* sequence *)

PARI

row(n) = if (n==0, [1], my(y=1/(1-x-x^2)); for (i=1, n, y = deriv(y)); (-1)^(n+1)*Vecrev(numerator(y/(n+1)!))); \\ Michel Marcus, Nov 27 2023