A327321 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.
1, 1, 3, 7, 18, 27, 5, 21, 27, 27, 61, 300, 630, 540, 405, 91, 549, 1350, 1890, 1215, 729, 547, 3822, 11529, 18900, 19845, 10206, 5103, 205, 1641, 5733, 11529, 14175, 11907, 5103, 2187, 4921, 44280, 177228, 412776, 622566, 612360, 428652, 157464, 59049, 7381
Offset: 1
Examples
p(x,3) = (1/k)((4 (7 + 18 x + 27 x^2))/(3 sqrt(3))), where k = 4/(3 sqrt(3)). First six rows: 1; 1, 3; 7, 18, 27; 5, 21, 27, 27; 61, 300, 630, 540, 405; 91, 549, 1350, 1890, 1215, 729; The first six polynomials, not factored: 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. The first six polynomials, factored: 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).
Programs
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Mathematica
c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[ MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@ Variables /@ #1 &)[List @@ poly], 0], poly]; r = Sqrt[3]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]]; Table[f[x, n], {n, 1, 6}] Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]] (* A327321 *) (* Peter J. C. Moses, Nov 01 2019 *)
Comments