cp's OEIS Frontend

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.

%I A363826 #13 Dec 23 2023 14:41:36
%S A363826 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 A363826 21,104,224,280,210,112,28,8,34,189,468,672,630,378,168,36,9,11,68,
%U A363826 189,312,336,252,126,48,9,2,89,605,1870,3465,4290,3696,2310,990
%N 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.
%C A363826 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 A363826 First eleven rows:
%e A363826    1
%e A363826    1    2
%e A363826    2    3    3
%e A363826    3    8    6    4
%e A363826    1    3    4    2    1
%e A363826    8   30   45   40   15    6
%e A363826   13   56  105  105   70   21    7
%e A363826   21  104  224  280  210  112   28   8
%e A363826   34  189  468  672  630  378  168  36   9
%e A363826   11   68  189  312  336  252  126  48   9  2
%e A363826   89  605 1870 3465 4290 3696 2310 990 330 55 11
%e A363826 Row 3 represents the polynomial 2 + 3*x + 3*x^2, extracted from
%e A363826 f"(x) = -((2*(2 + 3*x + 3*x^2))/(-1 + x + x^2)^3), where f(x) = 1/(1-x-x^2).
%t A363826 t = Table[CoefficientList[((-1)^(n + 1)) Numerator[Factor[D[1/(1 - x - x^2), {x, n}]]/(n + 1)!], x], {n, 0, 10}]
%t A363826 TableForm[t] (* array *)
%t A363826 Flatten[t ]  (* sequence *)
%o A363826 (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 A363826 Cf. A094440.
%K A363826 nonn,tabl
%O A363826 1,3
%A A363826 _Clark Kimberling_, Nov 26 2023