A070918 - OEIS

A070918

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

%I #21 Feb 12 2020 13:08:30

%S 1,-2,1,6,-5,1,-30,31,-10,1,210,-247,101,-17,1,-2310,2927,-1358,288,

%T -28,1,30030,-40361,20581,-5102,652,-41,1,-510510,716167,-390238,

%U 107315,-16186,1349,-58,1,9699690,-14117683,8130689,-2429223,414849,-41817,2451,-77,1

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

%C 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.

%H Alois P. Heinz, <a href="/A070918/b070918.txt">Rows n = 0..140, flattened</a>

%F From _Alois P. Heinz_, Aug 18 2019: (Start)

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

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

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

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

%e 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.

%e Triangle begins:

%e 1;

%e -2, 1;

%e 6, -5, 1;

%e -30, 31, -10, 1;

%e 210, -247, 101, -17, 1;

%e -2310, 2927, -1358, 288, -28, 1;

%e 30030, -40361, 20581, -5102, 652, -41, 1;

%e -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;

%e ...

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):

%p seq(T(n), n=0..10); # _Alois P. Heinz_, Aug 18 2019

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

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

%Y Cf. A008275 (Stirling numbers of first kind).

%Y Cf. A005867 (absolute values of row sums).

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

%Y Cf. A000040, A078456.

%K sign,tabl

%O 0,2

%A _Rick L. Shepherd_, May 20 2002

%E First term T(0,0)=1 prepended by _Alois P. Heinz_, Aug 18 2019