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A327318
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.
2
1, 3, 4, 7, 18, 12, 15, 56, 72, 32, 31, 150, 280, 240, 80, 63, 372, 900, 1120, 720, 192, 127, 882, 2604, 4200, 3920, 2016, 448, 255, 2032, 7056, 13888, 16800, 12544, 5376, 1024, 511, 4590, 18288, 42336, 62496, 60480, 37632, 13824, 2304, 1023, 10220, 45900
OFFSET
1,2
COMMENTS
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.
EXAMPLE
First six rows:
1;
3, 4;
7, 18, 12;
15, 56, 72, 32;
31, 150, 280, 240, 80;
63, 372, 900, 1120, 720, 192;
The first six polynomials, not factored:
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.
The first six polynomials, factored:
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).
MATHEMATICA
r = 1; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327318 array *)
Flatten[Table[c[x, n], {n, 1, 12}]] (* A327318 sequence *)
CROSSREFS
Cf. A327316, A327317, A000225 (x = 0), A005061 (x = 1), A081199 (x = 1/2).
Sequence in context: A078825 A361659 A331115 * A344783 A093611 A042375
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 08 2019
STATUS
approved