A279126 Triangle read by rows: coefficients of the polynomial P_n(x) of degree n-1 such that P_n(j) is the j-th prime == 1 (mod A000178(n-1)) for 1<=j<=n.

2, 1, 1, 1, 2, 0, 1, 2, 12, -2, 1729, -2712, 1980, -456, 36, -103679, 399744, -264960, 83520, -11520, 576, 4603392001, -10890547200, 9794649600, -4267987200, 962668800, -107308800, 4665600, 61326139392001, -149868925747200, 139442417049600, -65662061875200, 17248412160000

Offset

1,1

Comments

Given n, let p_1, p_2, ..., p_n be the first n primes == 1 (mod A000178(n-1)). Since A000178(n-1) is the determinant of the Vandermonde matrix of the numbers 1..n, the polynomial P(x) of degree <=n-1 with P(j) = p_j for 1<=j<=n has integer coefficients. Row n of the triangle consists of the coefficients of x^i in P(x) for i = 0,1,...,n-1.

Links

Robert Israel, Table of n, a(n) for n = 1..861 (rows 1 to 41, flattened)

Mathematics StackExchange, Can we always find some polynomial for which this holds?

Example

Triangle begins
2
1, 1
1, 2, 0
1, 2, 12, -2
1729, -2712, 1980, -456, 36
-103679, 399744, -264960, 83520, -11520, 576
4603392001, -10890547200, 9794649600, -4267987200, 962668800, -107308800, 4665600
61326139392001, -149868925747200, 139442417049600, -65662061875200, 17248412160000, -2552991436800, 198742118400, -6320332800
For n=4 the polynomial P(x) = 1+2x+12x^2-2x^3 has P(1)=13, P(2)=37, P(3)=61, P(4)=73, these values being the first four primes == 1 (mod 12).

Maple

with(LinearAlgebra):
for n from 1 to 10 do
  V:= VandermondeMatrix([$1..n]);
  d:= Determinant(V);
  S:= Vector(n);
  count:= 0;
  for k from 1 by d while count < n do
    if isprime(k) then
      count:= count+1;
      S[count]:= k
    fi
  od;
  R[n]:= convert(V^(-1) . S, list);
od:
seq(op(R[n[), n=1..10);

Mathematica

nmax = 10;
For[n = 1, n <= nmax, n++, V = Table[Range[n]^k, {k, 0, n-1}] // Transpose; d = Det[V]; S = Table[0, {n}]; count = 0; For[k = 1, count < n, k += d, If[PrimeQ[k], count++; S[[count]] = k]]; R[n] = Inverse[V].S];
Array[R, nmax] // Flatten (* Jean-François Alcover, Apr 29 2019 *)

Crossrefs

Cf. A000178.

Sequence in context: A088231 A327954 A336386 * A210679 A143262 A035180

Adjacent sequences: A279123 A279124 A279125 * A279127 A279128 A279129

Keyword

sign,tabl

Author

Robert Israel, Dec 08 2016

Status

approved