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A327317 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 = 2 and s = 1/2.

%I A327317 #10 Dec 15 2024 17:15:05
%S A327317 1,5,4,21,30,12,85,168,120,32,341,850,840,400,80,1365,4092,5100,3360,
%T A327317 1200,192,5461,19110,28644,23800,11760,3360,448,21845,87376,152880,
%U A327317 152768,95200,37632,8960,1024,87381,393210,786384,917280,687456,342720,112896
%N A327317 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 = 2 and s = 1/2.
%C A327317 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 A327317 First six rows:
%e A327317      1;
%e A327317      5,    4;
%e A327317     21,   30,   12;
%e A327317     85,  168,  120,   32;
%e A327317    341,  850,  840,  400,   80;
%e A327317   1365, 4092, 5100, 3360, 1200, 192;
%e A327317 The first six polynomials, not factored:
%e A327317 1, 5 + 4 x, 21 + 30 x + 12 x^2, 85 + 168 x + 120 x^2 + 32 x^3, 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 1365 + 4092 x + 5100 x^2 + 3360 x^3 + 1200 x^4 + 192 x^5.
%e A327317 The first six polynomials, factored:
%e A327317 1, 5 + 4 x, 3 (7 + 10 x + 4 x^2), (5 + 4 x) (17 + 20 x + 8 x^2), 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 3 (5 + 4 x) (7 + 10 x + 4 x^2) (13 + 10 x + 4 x^2).
%t A327317 r = 2; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
%t A327317 Column[Table[Expand[f[x, n]], {n, 1, 5}]]
%t A327317 c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
%t A327317 TableForm[Table[c[x, n], {n, 1, 10}]] (* A327317 array *)
%t A327317 Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327317 sequence *)
%Y A327317 Cf. A327316, A002450 (x=0), A016137 (x=1), A001045 (x = -1), A016162 (x = 2), A016181 (x = 3), A016127 (x = -3), A016157 (x = 1/2).
%K A327317 nonn,tabl
%O A327317 1,2
%A A327317 _Clark Kimberling_, Nov 03 2019