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%I #12 Apr 30 2020 15:23:46
%S 1,1,3,7,18,27,5,21,27,27,61,300,630,540,405,91,549,1350,1890,1215,
%T 729,547,3822,11529,18900,19845,10206,5103,205,1641,5733,11529,14175,
%U 11907,5103,2187,4921,44280,177228,412776,622566,612360,428652,157464,59049,7381
%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). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.
%e p(x,3) = (1/k)((4 (7 + 18 x + 27 x^2))/(3 sqrt(3))), where k = 4/(3 sqrt(3)).
%e First six rows:
%e 1;
%e 1, 3;
%e 7, 18, 27;
%e 5, 21, 27, 27;
%e 61, 300, 630, 540, 405;
%e 91, 549, 1350, 1890, 1215, 729;
%e The first six polynomials, not factored:
%e 1, 1 + 3 x, 7 + 18 x + 27 x^2, 5 + 21 x + 27 x^2 + 27 x^3, 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, 91 + 549 x + 1350 x^2 + 1890 x^3 + 1215 x^4 + 729 x^5.
%e The first six polynomials, factored:
%e 1, 1 + 3 x, 7 + 18 x + 27 x^2, (1 + 3 x) (5 + 6 x + 9 x^2), 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, (1 + 3 x) (13 + 6 x + 9 x^2) (7 + 18 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]; 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}]] (* A327321 *)
%t (* _Peter J. C. Moses_, Nov 01 2019 *)
%Y Cf. A327320, A329008, A329000, A031364.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Nov 08 2019
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