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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.
4

%I #6 Nov 06 2019 19:15:50

%S 1,1,6,7,9,27,13,84,54,108,11,39,126,54,81,133,990,1755,3780,1215,

%T 1458,463,2793,10395,12285,19845,5103,5103,1261,11112,33516,83160,

%U 73710,95256,20412,17496,4039,34047,150012,301644,561330,398034,428652,78732,59049

%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(3/2). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

%e We have p(x,3) = (1/k)((5 (7 + 9 x + 27 x^2))/(6 sqrt(6))), where k = 5/(6 sqrt(6)).

%e First six rows:

%e 1;

%e 1, 6;

%e 7, 9, 27;

%e 13, 84, 54, 108;

%e 11, 39, 126, 54, 81;

%e 133, 990, 1755, 3780, 1215, 1458;

%e The first six polynomials, not factored:

%e 1, 1 + 6 x, 7 + 9 x + 27 x^2, 13 + 84 x + 54 x^2 + 108 x^3, 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, 133 + 990 x + 1755 x^2 + 3780 x^3 + 1215 x^4 + 1458 x^5.

%e The first six polynomials, factored:

%e 1, 1 + 6 x, 7 + 9 x + 27 x^2, (1 + 6 x) (13 + 6 x + 18 x^2), 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, (1 + 6 x) (19 + 3 x + 9 x^2) (7 + 9 x + 27 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[3/2]; 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}]] (* A328644 *)

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

%Y Cf. A327320, A327321, A329017, A329018, A329019.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Nov 03 2019