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%I #10 Nov 11 2019 13:59:29
%S 1,3,4,7,18,12,15,56,72,32,31,150,280,240,80,63,372,900,1120,720,192,
%T 127,882,2604,4200,3920,2016,448,255,2032,7056,13888,16800,12544,5376,
%U 1024,511,4590,18288,42336,62496,60480,37632,13824,2304,1023,10220,45900
%N Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 1 and s = 1/2.
%C p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.
%e First six rows:
%e 1;
%e 3, 4;
%e 7, 18, 12;
%e 15, 56, 72, 32;
%e 31, 150, 280, 240, 80;
%e 63, 372, 900, 1120, 720, 192;
%e The first six polynomials, not factored:
%e 1, 3 + 4 x, 7 + 18 x + 12 x^2, 15 + 56 x + 72 x^2 + 32 x^3, 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, 63 + 372 x + 900 x^2 + 1120 x^3 + 720 x^4 + 192 x^5.
%e The first six polynomials, factored:
%e 1, 3 + 4 x, 7 + 18 x + 12 x^2, (3 + 4 x) (5 + 12 x + 8 x^2), 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, (3 + 4 x) (3 + 6 x + 4 x^2) (7 + 18 x + 12 x^2).
%t r = 1; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
%t Column[Table[Expand[f[x, n]], {n, 1, 5}]]
%t c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
%t TableForm[Table[c[x, n], {n, 1, 10}]] (* A327318 array *)
%t Flatten[Table[c[x, n], {n, 1, 12}]] (* A327318 sequence *)
%Y Cf. A327316, A327317, A000225 (x = 0), A005061 (x = 1), A081199 (x = 1/2).
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Nov 08 2019
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