A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

1, -2, 1, 6, -5, 1, -30, 31, -10, 1, 210, -247, 101, -17, 1, -2310, 2927, -1358, 288, -28, 1, 30030, -40361, 20581, -5102, 652, -41, 1, -510510, 716167, -390238, 107315, -16186, 1349, -58, 1, 9699690, -14117683, 8130689, -2429223, 414849, -41817, 2451, -77, 1

Offset: 0

Author

Rick L. Shepherd, May 20 2002

Keywords

• sign
• tabl

Comments

Analog of the Stirling numbers of the first kind (A008275): The Stirling numbers (beginning with the 2nd row) are the coefficients of the polynomials having exactly the first n natural numbers as roots. This sequence (beginning with first row) consists of the coefficients of the polynomials having exactly the first n prime numbers as roots.

Examples

			Row 4 of this sequence is 210, -247, 101, -17, 1 because (x-prime(1))(x-prime(2))(x-prime(3))(x-prime(4)) = (x-2)(x-3)(x-5)(x-7) = x^4 - 17*x^3 + 101*x^2 - 247*x + 210.
Triangle begins:
        1;
       -2,      1;
        6,     -5,       1;
      -30,     31,     -10,      1;
      210,   -247,     101,    -17,      1;
    -2310,   2927,   -1358,    288,    -28,    1;
    30030, -40361,   20581,  -5102,    652,  -41,   1;
  -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A008275 (Stirling numbers of first kind).

Cf. A005867 (absolute values of row sums).

Cf. A054640 (sum of absolute values of terms in rows).

Cf. A000040, A078456.

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

Mathematica

Table[CoefficientList[Expand[Times@@(x-Prime[Range[n]])],x],{n,0,10}]// Flatten (* Harvey P. Dale, Feb 12 2020 *)

PARI

p=1; for(k=1,10,p=p*(x-prime(k)); for(n=0,k,print1(polcoeff(p,n),",")))

Formula

From Alois P. Heinz, Aug 18 2019: (Start)

T(n,k) = [x^k] Product_{i=1..n} (x-prime(i)).

Sum_{k=0..n} |T(n,k)| = A054640(n).

|Sum_{k=0..n} T(n,k)| = A005867(n).

|Sum_{k=0..n} k * T(n,k)| = A078456(n). (End)

Extensions

First term T(0,0)=1 prepended by Alois P. Heinz, Aug 18 2019