A327322 - OEIS

%I #12 Nov 23 2019 13:39:45

%S 1,2,5,7,20,25,26,105,150,125,521,2600,5250,5000,3125,434,2605,6500,

%T 8750,6250,3125,13021,91140,273525,455000,459375,262500,109375,8138,

%U 65105,227850,455875,568750,459375,218750,78125,36169,325520,1302100,3038000,4558750

%N Triangular array read by rows: row n shows the coefficients of the polynomial p(x,n) constructed as in Comments; these polynomials form a strong divisibility sequence.

%C Suppose q is a rational number such that the number r = sqrt(q) is irrational. The function (r x + r)^n - (r x - 1/r)^n of x can be represented as k*p(x,n), where k is a constant and p(x,n) is a product of nonconstant polynomials having gcd = 1; the sequence p(x,n) is a strong divisibility sequence of polynomials; i.e., gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)).  For A327320, r = sqrt(5).  If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

%e p(x,3) = (1/k)((18 (7 + 20 x + 25 x^2))/(5 sqrt(5))), where k = 18/(5 sqrt(5)).

%e First six rows:

%e     1;

%e     2,    5;

%e     7,   20,   25;

%e    26,  105,  150,  125;

%e   521, 2600, 5250, 5000, 3125;

%e   434, 2605, 6500, 8750, 6250, 3125;

%e The first six polynomials, not factored:

%e 1, 2 + 5 x, 7 + 20 x + 25 x^2, 26 + 105 x + 150 x^2 + 125 x^3, 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, 434 + 2605 x + 6500 x^2 + 8750 x^3 + 6250 x^4 + 3125 x^5.

%e The first six polynomials, factored:

%e 1, 2 + 5 x, 7 + 20 x + 25 x^2, (2 + 5 x) (13 + 20 x + 25 x^2), 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, (2 + 5 x) (7 + 20 x + 25 x^2) (31 + 20 x + 25 x^2).

%t c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[

%t MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@

%t Variables /@ #1 &)[List @@ poly], 0], poly];

%t r = Sqrt[5]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];

%t Table[f[x, n], {n, 1, 6}]

%t Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]  (* A327322 *)

%t (* _Peter J. C. Moses_, Nov 01 2019 *)

%Y Cf. A327320, A327321.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Nov 08 2019