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

1,1

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.

Links

Alois P. Heinz, Rows n = 1..141, flattened

Fedor Igumnov, T(n,k) for n = 1..50

Example

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

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]]
]

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.

Sequence in context: A236248 A073825 A015891 * A160645 A248616 A341522

Adjacent sequences: A238143 A238144 A238145 * A238147 A238148 A238149

Keyword

sign,easy,tabl

Author

Fedor Igumnov, Feb 18 2014

Extensions

Name edited by Alois P. Heinz, Aug 18 2019

Status

approved