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A327316
Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = ((x+r)^n - (x+s)^n)/(r - s), where r = 3 and s = 2.
3
1, 5, 2, 19, 15, 3, 65, 76, 30, 4, 211, 325, 190, 50, 5, 665, 1266, 975, 380, 75, 6, 2059, 4655, 4431, 2275, 665, 105, 7, 6305, 16472, 18620, 11816, 4550, 1064, 140, 8, 19171, 56745, 74124, 55860, 26586, 8190, 1596, 180, 9, 58025, 191710, 283725, 247080
OFFSET
1,2
COMMENTS
For every choice of integers r and s, the polynomials p(n,x) form a strong divisibility sequence. Thus, if r, s, and x are integers, then p(x,n) is a strong divisibility sequence. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)).
EXAMPLE
First seven rows:
1
5 2
19 15 3
65 76 30 4
211 325 190 50 5
665 1266 975 380 75 6
2059 4655 4431 2275 665 105 7
MATHEMATICA
f[x_, n_] := ((x + r)^n - (x + s)^n)/(r - s);
r = 3; s = 2;
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}]] (* A327316 array *)
Flatten[Table[c[x, n], {n, 1, 12}]] (* A327316 sequence *)
CROSSREFS
Cf. A001047 (x=0), A005061 (x=1), A005060 (x=2), A005062 (x=3), A081200 (x=1/2).
Sequence in context: A304635 A356330 A306198 * A206582 A276533 A303685
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 01 2019
STATUS
approved