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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. 6
-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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
sign,easy,tabl
AUTHOR
Fedor Igumnov, Feb 18 2014
EXTENSIONS
Name edited by Alois P. Heinz, Aug 18 2019
STATUS
approved

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Last modified July 22 00:58 EDT 2024. Contains 374478 sequences. (Running on oeis4.)