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Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.
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%I #49 Sep 08 2022 08:46:03

%S 1,1,0,1,1,0,1,0,1,0,1,-1,-1,1,0,1,0,0,0,1,0,1,-1,1,1,-1,1,0,1,0,-1,0,

%T -1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,-1,0,1,0,1,1,-1,1,1,1,1,-1,1,

%U 1,0,1,0,0,0,-1,0,-1,0,0,0,1,0

%N Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.

%C Signed version of A054521.

%D Henri Cohen: A Course in Computational Algebraic Number Theory, p. 29.

%H G. C. Greubel, <a href="/A215200/b215200.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>.

%e Triangle begins:

%e 1,

%e 1, 0,

%e 1, 1, 0,

%e 1, 0, 1, 0,

%e 1, -1, -1, 1, 0,

%e 1, 0, 0, 0, 1, 0,

%e 1, -1, 1, 1, -1, 1, 0,

%e 1, 0, -1, 0, -1, 0, 1, 0,

%e 1, 1, 0, 1, 1, 0, 1, 1, 0,

%e 1, 0, 1, 0, 0, 0, -1, 0, 1, 0,

%e From _Jianing Song_, Dec 26 2018: (Start)

%e This sequence can also be arranged into a square array T(n,k) = Kronecker symbol(n|k) with n >= 0, k >= 1, read by antidiagonals:

%e 1 0 0 0 0 0 0 ... ((0|k) = A000007(k+1))

%e 1 1 1 1 1 1 1 ... ((1|k) = A000012)

%e 1 0 -1 0 -1 0 -1 ... ((2|k) = A091337)

%e 1 -1 0 1 -1 0 -1 ... ((3|k) = A091338)

%e 1 0 1 0 1 0 1 ... ((4|k) = A000035)

%e 1 -1 -1 1 0 1 -1 ... ((5|k) = A080891)

%e 1 0 0 0 1 0 -1 ... ((6|k) = A322796)

%e 1 1 1 1 -1 1 0 ... ((7|k) = A089509)

%e ... (End)

%p A215200_row := n -> seq(numtheory[jacobi](n-k,k),k=1..n);

%p for n from 1 to 13 do A215200_row(n) od;

%t Column[Table[KroneckerSymbol[n - k, k], {n, 10}, {k, n}], Center] (* _Alonso del Arte_, Aug 06 2012 *)

%o (Sage)

%o def A215200_row(n): return [kronecker_symbol(n-k,k) for k in (1..n)]

%o for n in (1..13): print(A215200_row(n))

%o (PARI) T(n,k) = kronecker(n-k, k);

%o tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Apr 24 2018

%o (Magma) /* As triangle */ [[KroneckerSymbol(n-k, k): k in [1..n]]: n in [1..21]]; // _Vincenzo Librandi_, Apr 24 2018

%Y Rows of square array include: A000012, A091337, A091338, A000035, A080891, A322796, A089509.

%Y Cf. A054521, A215283, A215284, A215285.

%K sign,tabl

%O 1,1

%A _Peter Luschny_, Aug 05 2012