Fedor Igumnov - cp's OEIS frontend

User: Fedor Igumnov

Fedor Igumnov has authored 2 sequences.

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)) = 1x^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

Fedor Igumnov, Feb 18 2014

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.

			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

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

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.

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

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

Name edited by Alois P. Heinz, Aug 18 2019

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376

Offset: 1

Fedor Igumnov, Jan 28 2014

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Fedor Igumnov, T(n,k) for n = 1..26

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

/* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014

t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

T(n,k) = T(n-1,k) + T(n-1,k+1).

Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

More terms from Bruno Berselli, Jan 28 2014

cp's OEIS Frontend

User: Fedor Igumnov

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)) = 1x^n + T(n,1)x^(n-1)+ ... + T(n,k-1)*x + T(n,k), for 1 <= k <= n.

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A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

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cp's OEIS Frontend

User: Fedor Igumnov

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.

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Programs

Maple

Mathematica

Extensions

A236538 Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Magma

Mathematica

Formula

Extensions

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)) = 1x^n + T(n,1)x^(n-1)+ ... + T(n,k-1)*x + T(n,k), for 1 <= k <= n.

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.