This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327322 #12 Nov 23 2019 13:39:45
%S A327322 1,2,5,7,20,25,26,105,150,125,521,2600,5250,5000,3125,434,2605,6500,
%T A327322 8750,6250,3125,13021,91140,273525,455000,459375,262500,109375,8138,
%U A327322 65105,227850,455875,568750,459375,218750,78125,36169,325520,1302100,3038000,4558750
%N A327322 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 A327322 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(5). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.
%e A327322 p(x,3) = (1/k)((18 (7 + 20 x + 25 x^2))/(5 sqrt(5))), where k = 18/(5 sqrt(5)).
%e A327322 First six rows:
%e A327322 1;
%e A327322 2, 5;
%e A327322 7, 20, 25;
%e A327322 26, 105, 150, 125;
%e A327322 521, 2600, 5250, 5000, 3125;
%e A327322 434, 2605, 6500, 8750, 6250, 3125;
%e A327322 The first six polynomials, not factored:
%e A327322 1, 2 + 5 x, 7 + 20 x + 25 x^2, 26 + 105 x + 150 x^2 + 125 x^3, 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, 434 + 2605 x + 6500 x^2 + 8750 x^3 + 6250 x^4 + 3125 x^5.
%e A327322 The first six polynomials, factored:
%e A327322 1, 2 + 5 x, 7 + 20 x + 25 x^2, (2 + 5 x) (13 + 20 x + 25 x^2), 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, (2 + 5 x) (7 + 20 x + 25 x^2) (31 + 20 x + 25 x^2).
%t A327322 c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[
%t A327322 MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@
%t A327322 Variables /@ #1 &)[List @@ poly], 0], poly];
%t A327322 r = Sqrt[5]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
%t A327322 Table[f[x, n], {n, 1, 6}]
%t A327322 Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]] (* A327322 *)
%t A327322 (* _Peter J. C. Moses_, Nov 01 2019 *)
%Y A327322 Cf. A327320, A327321.
%K A327322 nonn,tabl
%O A327322 1,2
%A A327322 _Clark Kimberling_, Nov 08 2019