A363826 - OEIS

%I #13 Dec 23 2023 14:41:36

%S 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,

%T 21,104,224,280,210,112,28,8,34,189,468,672,630,378,168,36,9,11,68,

%U 189,312,336,252,126,48,9,2,89,605,1870,3465,4290,3696,2310,990

%N 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.

%C 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.

%e First eleven rows:

%e    1

%e    1    2

%e    2    3    3

%e    3    8    6    4

%e    1    3    4    2    1

%e    8   30   45   40   15    6

%e   13   56  105  105   70   21    7

%e   21  104  224  280  210  112   28   8

%e   34  189  468  672  630  378  168  36   9

%e   11   68  189  312  336  252  126  48   9  2

%e   89  605 1870 3465 4290 3696 2310 990 330 55 11

%e Row 3 represents the polynomial 2 + 3*x + 3*x^2, extracted from

%e f"(x) = -((2*(2 + 3*x + 3*x^2))/(-1 + x + x^2)^3), where f(x) = 1/(1-x-x^2).

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

%t TableForm[t] (* array *)

%t Flatten[t ]  (* sequence *)

%o (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

%Y Cf. A094440.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Nov 26 2023