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A327320 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.

%I A327320 #10 Nov 23 2019 13:39:18
%S A327320 1,1,4,1,2,4,5,24,24,32,11,50,120,80,80,7,44,100,160,80,64,43,294,924,
%T A327320 1400,1680,672,448,85,688,2352,4928,5600,5376,1792,1024,19,170,688,
%U A327320 1568,2464,2240,1792,512,256,341,3420,15300,41280,70560,88704,67200
%N A327320 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 A327320 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(2).  If x is an integer, then p(x,n) is a strong divisibility sequence of integers.
%e A327320 p(x,3) = (1/k)(9 (1 + 2 x + 4 x^2))/(2 sqrt(2)), where k = 9/(2 sqrt(2)).
%e A327320 First six rows:
%e A327320    1;
%e A327320    1,  4;
%e A327320    1,  2,   4;
%e A327320    5, 24,  24,  32;
%e A327320   11, 50, 120,  80, 80;
%e A327320   7,  44, 100, 160, 80, 64;
%e A327320 The first six polynomials, not factored:
%e A327320 1, 1 + 4 x, 1 + 2 x + 4 x^2, 5 + 24 x + 24 x^2 + 32 x^3, 11 + 50 x + 120 x^2 + 80 x^3 + 80 x^4, 7 + 44 x + 100 x^2 + 160 x^3 + 80 x^4 + 64 x^5.
%e A327320 The first six polynomials, factored:
%e A327320 1, 1 + 4 x, 1 + 2 x + 4 x^2, (1 + 4 x) (5 + 4 x + 8 x^2), 11 + 50 x + 120 x^2 + 80 x^3 + 80 x^4, (1 + 4 x) (1 + 2 x + 4 x^2) (7 + 2 x + 4 x^2).
%t A327320 c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[
%t A327320 MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@
%t A327320 Variables /@ #1 &)[List @@ poly], 0], poly];
%t A327320 r = Sqrt[2]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
%t A327320 Table[f[x, n], {n, 1, 6}]
%t A327320 Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]  (* A327320 *)
%t A327320 (* _Peter J. C. Moses_, Nov 01 2019 *)
%Y A327320 Cf. A327315.
%K A327320 nonn,tabl
%O A327320 1,3
%A A327320 _Clark Kimberling_, Nov 08 2019