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A215200 Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.

%I A215200 #50 Feb 16 2025 08:33:18
%S A215200 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 A215200 -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 A215200 1,0,1,0,0,0,-1,0,-1,0,0,0,1,0
%N A215200 Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.
%C A215200 Signed version of A054521.
%D A215200 Henri Cohen: A Course in Computational Algebraic Number Theory, p. 29.
%H A215200 G. C. Greubel, <a href="/A215200/b215200.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H A215200 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>.
%e A215200 Triangle begins:
%e A215200   1,
%e A215200   1,  0,
%e A215200   1,  1,  0,
%e A215200   1,  0,  1, 0,
%e A215200   1, -1, -1, 1,  0,
%e A215200   1,  0,  0, 0,  1, 0,
%e A215200   1, -1,  1, 1, -1, 1,  0,
%e A215200   1,  0, -1, 0, -1, 0,  1, 0,
%e A215200   1,  1,  0, 1,  1, 0,  1, 1, 0,
%e A215200   1,  0,  1, 0,  0, 0, -1, 0, 1, 0,
%e A215200 From _Jianing Song_, Dec 26 2018: (Start)
%e A215200 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 A215200   1  0  0  0  0  0  0 ... ((0|k) = A000007(k+1))
%e A215200   1  1  1  1  1  1  1 ... ((1|k) = A000012)
%e A215200   1  0 -1  0 -1  0 -1 ... ((2|k) = A091337)
%e A215200   1 -1  0  1 -1  0 -1 ... ((3|k) = A091338)
%e A215200   1  0  1  0  1  0  1 ... ((4|k) = A000035)
%e A215200   1 -1 -1  1  0  1 -1 ... ((5|k) = A080891)
%e A215200   1  0  0  0  1  0 -1 ... ((6|k) = A322796)
%e A215200   1  1  1  1 -1  1  0 ... ((7|k) = A089509)
%e A215200   ... (End)
%p A215200 A215200_row := n -> seq(numtheory[jacobi](n-k,k),k=1..n);
%p A215200 for n from 1 to 13 do A215200_row(n) od;
%t A215200 Column[Table[KroneckerSymbol[n - k, k], {n, 10}, {k, n}], Center] (* _Alonso del Arte_, Aug 06 2012 *)
%o A215200 (Sage)
%o A215200 def A215200_row(n): return [kronecker_symbol(n-k,k) for k in (1..n)]
%o A215200 for n in (1..13): print(A215200_row(n))
%o A215200 (PARI) T(n,k) = kronecker(n-k, k);
%o A215200 tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Apr 24 2018
%o A215200 (Magma) /* As triangle */ [[KroneckerSymbol(n-k, k):  k in [1..n]]: n in [1..21]]; // _Vincenzo Librandi_, Apr 24 2018
%Y A215200 Rows of square array include: A000012, A091337, A091338, A000035, A080891, A322796, A089509.
%Y A215200 Cf. A054521, A215283, A215284, A215285.
%K A215200 sign,tabl
%O A215200 1,1
%A A215200 _Peter Luschny_, Aug 05 2012