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

0,2

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.

Links

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

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)

Example

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;
  ...

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 *)

Prog

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

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.

Sequence in context: A143491 A308498 A295517 * A113381 A228175 A118980

Adjacent sequences: A070915 A070916 A070917 * A070919 A070920 A070921

Keyword

sign,tabl

Author

Rick L. Shepherd, May 20 2002

Extensions

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

Status

approved