A327316 - OEIS

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.

%I #12 Nov 07 2019 04:24:55

%S 1,5,2,19,15,3,65,76,30,4,211,325,190,50,5,665,1266,975,380,75,6,2059,

%T 4655,4431,2275,665,105,7,6305,16472,18620,11816,4550,1064,140,8,

%U 19171,56745,74124,55860,26586,8190,1596,180,9,58025,191710,283725,247080

%N 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.

%C 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)).

%e First seven rows:

%e 1

%e 5 2

%e 19 15 3

%e 65 76 30 4

%e 211 325 190 50 5

%e 665 1266 975 380 75 6

%e 2059 4655 4431 2275 665 105 7

%t f[x_, n_] := ((x + r)^n - (x + s)^n)/(r - s);

%t r = 3; s = 2;

%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}]] (* A327316 array *)

%t Flatten[Table[c[x, n], {n, 1, 12}]] (* A327316 sequence *)

%Y Cf. A001047 (x=0), A005061 (x=1), A005060 (x=2), A005062 (x=3), A081200 (x=1/2).

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Nov 01 2019