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.

-2, -5, 6, -10, 31, -30, -17, 101, -247, 210, -28, 288, -1358, 2927, -2310, -41, 652, -5102, 20581, -40361, 30030, -58, 1349, -16186, 107315, -390238, 716167, -510510, -77, 2451, -41817, 414849, -2429223, 8130689, -14117683, 9699690

Offset: 1

Author

Fedor Igumnov, Feb 18 2014

Keywords

• sign
• easy
• tabl

Comments

The coefficient of first polynomial term with highest degree is always 1.

Each number in triangle is the sum of radicals of integers.

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.

Examples

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |  -2;
2  |  -5,   6;
3  |  -10,  31,   -30;
4  |  -17, 101,  -247,  210;
5  |  -28, 288, -1358,  2927,  -2310;
6  |  -41, 652, -5102, 20581, -40361, 30030;
7  |  -58,1349,-16186,107315,-390238,716167,-510510;
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
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Fedor Igumnov, T(n,k) for n = 1..50

Crossrefs

Cf. A007504 (abs of column 1) A002110(abs of right border). Also:

A024447 is the abs of column 2;

A024448 is the abs of column 3;

A024449 is the abs of column 4;

A006939 is the determinant of triangle matrix, considering T(n,k) k>n = 0;

A007947 = radicals of integers.

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, n-i), i=1..n))(mul(x-ithprime(i), i=1..n)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 18 2019

Mathematica

a = 1
For [i = 1, i < 10, i++,
a *= (x - Prime[i]);
Print[Drop[Reverse[CoefficientList[Expand[a], x]], 1]]
]

Extensions

Name edited by Alois P. Heinz, Aug 18 2019