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A238146 Triangle read by rows: T(n,k) is coefficient of x^(n-k) in consecutive prime rooted polynomial of degree n, P(x) = Product_{k=1..n} (x-p(k)) = 1*x^n + T(n,1)*x^(n-1)+ ... + T(n,k-1)*x + T(n,k), for 1 <= k <= n.

%I A238146 #21 Aug 18 2019 17:10:12
%S A238146 -2,-5,6,-10,31,-30,-17,101,-247,210,-28,288,-1358,2927,-2310,-41,652,
%T A238146 -5102,20581,-40361,30030,-58,1349,-16186,107315,-390238,716167,
%U A238146 -510510,-77,2451,-41817,414849,-2429223,8130689,-14117683,9699690
%N A238146 Triangle read by rows: T(n,k) is coefficient of x^(n-k) in consecutive prime rooted polynomial of degree n, P(x) = Product_{k=1..n} (x-p(k)) = 1*x^n + T(n,1)*x^(n-1)+ ... + T(n,k-1)*x + T(n,k), for 1 <= k <= n.
%C A238146 The coefficient of first polynomial term with highest degree is always 1.
%C A238146 Each number in triangle is the sum of radicals of integers.
%C A238146 The absolute value of the entry in the k-th column is the k-th elementary symmetric function of the first n+(k-1) primes.
%H A238146 Alois P. Heinz, <a href="/A238146/b238146.txt">Rows n = 1..141, flattened</a>
%H A238146 Fedor Igumnov, <a href="/A238146/a238146.txt">T(n,k) for n = 1..50</a>
%e A238146 Triangle begins:
%e A238146 ================================================
%e A238146 \k |    1     2     3     4     5     6     7
%e A238146 n\ |
%e A238146 ================================================
%e A238146 1  |  -2;
%e A238146 2  |  -5,   6;
%e A238146 3  |  -10,  31,   -30;
%e A238146 4  |  -17, 101,  -247,  210;
%e A238146 5  |  -28, 288, -1358,  2927,  -2310;
%e A238146 6  |  -41, 652, -5102, 20581, -40361, 30030;
%e A238146 7  |  -58,1349,-16186,107315,-390238,716167,-510510;
%e A238146 So equation x^7 -58*x^6 + 1349*x^5 -16186*x^4 + 107315*x^3 -390238*x^2+ 716167*x -510510 = 0 has 7 consecutive prime roots: 2,3,5,7,11,13,17
%p A238146 T:= n-> (p-> seq(coeff(p, x, n-i), i=1..n))(mul(x-ithprime(i), i=1..n)):
%p A238146 seq(T(n), n=1..10);  # _Alois P. Heinz_, Aug 18 2019
%t A238146 a = 1
%t A238146 For [i = 1, i < 10, i++,
%t A238146 a *= (x - Prime[i]);
%t A238146 Print[Drop[Reverse[CoefficientList[Expand[a], x]], 1]]
%t A238146 ]
%Y A238146 Cf. A007504 (abs of column 1) A002110(abs of right border). Also:
%Y A238146 A024447 is the abs of column 2;
%Y A238146 A024448 is the abs of column 3;
%Y A238146 A024449 is the abs of column 4;
%Y A238146 A006939 is the determinant of triangle matrix, considering T(n,k) k>n = 0;
%Y A238146 A007947 = radicals of integers.
%K A238146 sign,easy,tabl
%O A238146 1,1
%A A238146 _Fedor Igumnov_, Feb 18 2014
%E A238146 Name edited by _Alois P. Heinz_, Aug 18 2019