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A327323 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 A327323 #8 Nov 23 2019 13:39:56
%S A327323 1,5,12,31,90,108,185,744,1080,864,1111,5550,11160,10800,6480,6665,
%T A327323 39996,99900,133920,97200,46656,5713,39990,119988,199800,200880,
%U A327323 116640,46656,239945,1919568,6718320,13438656,16783200,13499136,6531840,2239488,1439671
%N A327323 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 A327323 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(6).  If x is an integer, then p(x,n) is a strong divisibility sequence of integers.
%e A327323 p(x,3) = (1/k)((7 (31 + 90 x + 108 x^2))/(6 sqrt(6))), where k = 7/(6 sqrt(6)).
%e A327323 First six rows:
%e A327323      1;
%e A327323      5,    12;
%e A327323     31,    90,    108;
%e A327323    185,   744,   1080,    864;
%e A327323   1111,  5550,  11160,  10800,   6480;
%e A327323   6665, 39996,  99900, 133920,  97200,  46656;
%e A327323   5713, 39990, 119988, 199800, 200880, 116640, 46656;
%e A327323 The first six polynomials, not factored:
%e A327323 1, 5 + 12 x, 31 + 90 x + 108 x^2, 185 + 744 x + 1080 x^2 + 864 x^3, 1111 + 5550 x + 11160 x^2 + 10800 x^3 + 6480 x^4, 6665 + 39996 x + 99900 x^2 + 133920 x^3 + 97200 x^4 + 46656 x^5.
%e A327323 The first six polynomials, factored:
%e A327323 1, 5 + 12 x, 31 + 90 x + 108 x^2, (5 + 12 x) (37 + 60 x + 72 x^2), 1111 + 5550 x +  11160 x^2 + 10800 x^3 + 6480 x^4, (5 + 12 x) (43 + 30 x + 36 x^2) (31 + 90 x + 108 x^2).
%t A327323 c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[
%t A327323 MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@
%t A327323 Variables /@ #1 &)[List @@ poly], 0], poly];
%t A327323 r = Sqrt[6]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
%t A327323 Table[f[x, n], {n, 1, 6}]
%t A327323 Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]  (* A327323 *)
%t A327323 (* _Peter J. C. Moses_, Nov 01 2019 *)
%Y A327323 Cf. A327320, A327321, A327322, A329014, A329015, A329016.
%K A327323 nonn,tabl
%O A327323 1,2
%A A327323 _Clark Kimberling_, Nov 09 2019