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

%I A070918 #21 Feb 12 2020 13:08:30
%S A070918 1,-2,1,6,-5,1,-30,31,-10,1,210,-247,101,-17,1,-2310,2927,-1358,288,
%T A070918 -28,1,30030,-40361,20581,-5102,652,-41,1,-510510,716167,-390238,
%U A070918 107315,-16186,1349,-58,1,9699690,-14117683,8130689,-2429223,414849,-41817,2451,-77,1
%N A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.
%C A070918 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 A070918 Alois P. Heinz, <a href="/A070918/b070918.txt">Rows n = 0..140, flattened</a>
%F A070918 From _Alois P. Heinz_, Aug 18 2019: (Start)
%F A070918 T(n,k) = [x^k] Product_{i=1..n} (x-prime(i)).
%F A070918 Sum_{k=0..n} |T(n,k)| = A054640(n).
%F A070918 |Sum_{k=0..n} T(n,k)| = A005867(n).
%F A070918 |Sum_{k=0..n} k * T(n,k)| = A078456(n). (End)
%e A070918 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 A070918 Triangle begins:
%e A070918         1;
%e A070918        -2,      1;
%e A070918         6,     -5,       1;
%e A070918       -30,     31,     -10,      1;
%e A070918       210,   -247,     101,    -17,      1;
%e A070918     -2310,   2927,   -1358,    288,    -28,    1;
%e A070918     30030, -40361,   20581,  -5102,    652,  -41,   1;
%e A070918   -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;
%e A070918   ...
%p A070918 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):
%p A070918 seq(T(n), n=0..10);  # _Alois P. Heinz_, Aug 18 2019
%t A070918 Table[CoefficientList[Expand[Times@@(x-Prime[Range[n]])],x],{n,0,10}]// Flatten (* _Harvey P. Dale_, Feb 12 2020 *)
%o A070918 (PARI) p=1; for(k=1,10,p=p*(x-prime(k)); for(n=0,k,print1(polcoeff(p,n),",")))
%Y A070918 Cf. A008275 (Stirling numbers of first kind).
%Y A070918 Cf. A005867 (absolute values of row sums).
%Y A070918 Cf. A054640 (sum of absolute values of terms in rows).
%Y A070918 Cf. A000040, A078456.
%K A070918 sign,tabl
%O A070918 0,2
%A A070918 _Rick L. Shepherd_, May 20 2002
%E A070918 First term T(0,0)=1 prepended by _Alois P. Heinz_, Aug 18 2019