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
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