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 A327323 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
 1, 5, 12, 31, 90, 108, 185, 744, 1080, 864, 1111, 5550, 11160, 10800, 6480, 6665, 39996, 99900, 133920, 97200, 46656, 5713, 39990, 119988, 199800, 200880, 116640, 46656, 239945, 1919568, 6718320, 13438656, 16783200, 13499136, 6531840, 2239488, 1439671 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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(6).  If x is an integer, then p(x,n) is a strong divisibility sequence of integers. LINKS EXAMPLE p(x,3) = (1/k)((7 (31 + 90 x + 108 x^2))/(6 sqrt(6))), where k = 7/(6 sqrt(6)). First six rows:      1;      5,    12;     31,    90,    108;    185,   744,   1080,    864;   1111,  5550,  11160,  10800,   6480;   6665, 39996,  99900, 133920,  97200,  46656;   5713, 39990, 119988, 199800, 200880, 116640, 46656; The first six polynomials, not factored: 1, 5 + 12 x, 31 + 90 x + 108 x^2, 185 + 744 x + 1080 x^2 + 864 x^3, 1111 + 5550 x + 11160 x^2 + 10800 x^3 + 6480 x^4, 6665 + 39996 x + 99900 x^2 + 133920 x^3 + 97200 x^4 + 46656 x^5. The first six polynomials, factored: 1, 5 + 12 x, 31 + 90 x + 108 x^2, (5 + 12 x) (37 + 60 x + 72 x^2), 1111 + 5550 x +  11160 x^2 + 10800 x^3 + 6480 x^4, (5 + 12 x) (43 + 30 x + 36 x^2) (31 + 90 x + 108 x^2). 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; 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}]]  (* A327323 *) (* Peter J. C. Moses, Nov 01 2019 *) CROSSREFS Cf. A327320, A327321, A327322, A329014, A329015, A329016. Sequence in context: A038357 A090974 A038606 * A192303 A301785 A066280 Adjacent sequences:  A327320 A327321 A327322 * A327324 A327325 A327326 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Nov 09 2019 STATUS approved

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Last modified August 11 03:41 EDT 2022. Contains 356046 sequences. (Running on oeis4.)