# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a328644 Showing 1-1 of 1 %I A328644 #6 Nov 06 2019 19:15:50 %S A328644 1,1,6,7,9,27,13,84,54,108,11,39,126,54,81,133,990,1755,3780,1215, %T A328644 1458,463,2793,10395,12285,19845,5103,5103,1261,11112,33516,83160, %U A328644 73710,95256,20412,17496,4039,34047,150012,301644,561330,398034,428652,78732,59049 %N A328644 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 A328644 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 A328644 We have p(x,3) = (1/k)((5 (7 + 9 x + 27 x^2))/(6 sqrt(6))), where k = 5/(6 sqrt(6)). %e A328644 First six rows: %e A328644 1; %e A328644 1, 6; %e A328644 7, 9, 27; %e A328644 13, 84, 54, 108; %e A328644 11, 39, 126, 54, 81; %e A328644 133, 990, 1755, 3780, 1215, 1458; %e A328644 The first six polynomials, not factored: %e A328644 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 A328644 The first six polynomials, factored: %e A328644 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 A328644 c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[ %t A328644 MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@ %t A328644 Variables /@ #1 &)[List @@ poly], 0], poly]; %t A328644 r = Sqrt[3/2]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]]; %t A328644 Table[f[x, n], {n, 1, 6}] %t A328644 Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]] (* A328644 *) %t A328644 (* _Peter J. C. Moses_, Nov 01 2019 *) %Y A328644 Cf. A327320, A327321, A329017, A329018, A329019. %K A328644 nonn,tabl %O A328644 1,3 %A A328644 _Clark Kimberling_, Nov 03 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE