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.

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)).

Links

Table of n, a(n) for n=1..49.

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

Adjacent sequences: A327313 A327314 A327315 * A327317 A327318 A327319

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 01 2019

Status

approved