A327855 Irregular triangle read by rows: T(n,k) = [x^k] (Sum_{i=0..prime(n)-1} (1+Legendre(i,prime(n))) * x^i)^2, for 0 <= k <= 2*prime(n)-2.

1, 4, 4, 1, 4, 4, 0, 0, 1, 4, 4, 0, 4, 8, 0, 0, 4, 1, 4, 8, 8, 8, 8, 8, 0, 4, 0, 0, 0, 0, 1, 4, 4, 4, 12, 12, 12, 8, 12, 12, 12, 0, 8, 8, 8, 0, 0, 0, 4, 0, 0, 1, 4, 4, 4, 12, 8, 4, 8, 4, 4, 12, 8, 12, 24, 8, 8, 8, 0, 4, 8, 4, 8, 8, 0, 4

Offset

1,2

Links

Table of n, a(n) for n=1..76.

Wikipedia, Gauss sum.

Wikipedia, Legendre symbol.

Formula

For n > 1, Sum_{k=0..2*prime(n)-2} T(n,k)*x^k == (-1)^((p - 1)/2) * p mod ((x^p - 1)/(x - 1)) where p is n-th prime.

Example

Triangle begins
[1, 4, 4],
[1, 4, 4, 0,  0],
[1, 4, 4, 0,  4,  8,  0, 0,  4],
[1, 4, 8, 8,  8,  8,  8, 0,  4,  0,  0, 0,  0],
[1, 4, 4, 4, 12, 12, 12, 8, 12, 12, 12, 0,  8,  8, 8, 0, 0, 0, 4, 0, 0],
[1, 4, 4, 4, 12,  8,  4, 8,  4,  4, 12, 8, 12, 24, 8, 8, 8, 0, 4, 8, 4, 8, 8, 0, 4],
...
------------------------------------------
1 + 4*x + 4*x^2 = 4*(x^3 - 1)/(x - 1) - 3.
1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 = 4 * (x^4 - x^3 + 2*x - 1)*(x^5 - 1)/(x - 1) + 5.
1 + 4*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 8*x^6 + 4*x^8 = 4 * (x^2 - x + 2)*(x^7 - 1)/(x - 1) - 7.

Prog

(PARI) forprime(p=2, 30, print(Vecrev((sum(k=0, p-1, (1+kronecker(k, p))*x^k))^2, 2*p-1), ", "))

Crossrefs

Cf. A073579, A226520.

Sequence in context: A030788 A087709 A106642 * A213056 A135012 A156380

Adjacent sequences: A327852 A327853 A327854 * A327856 A327857 A327858

Keyword

nonn,tabf

Author

Seiichi Manyama, Sep 28 2019

Status

approved