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A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

%I A226519 #14 Oct 06 2024 09:15:52
%S A226519 1,1,0,1,0,-1,0,1,2,1,2,1,0,1,0,1,2,3,2,1,0,1,0,1,0,1,2,1,0,-1,-2,-1,
%T A226519 0,-1,0,1,2,1,2,1,0,-1,0,1,0,-1,-2,-1,-2,-1,0,1,0,-1,0,1,2,3,2,3,2,3,
%U A226519 2,1,0,-1,0,1,0,1,2,3,4,3,4,3,4,5,4,3,4,5,4,3,4,3,4,3,2,1,0
%N A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).
%C A226519 Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.
%D A226519 József Beck, Inevitable randomness in discrete mathematics, University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.
%H A226519 G. C. Greubel, <a href="/A226519/b226519.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%e A226519 Triangle begins:
%e A226519   1;
%e A226519   1, 0;
%e A226519   1, 0, -1, 0;
%e A226519   1, 2,  1, 2, 1, 0;
%e A226519   1, 0,  1, 2, 3, 2,  1,  0,  1, 0;
%e A226519   1, 0,  1, 2, 1, 0, -1, -2, -1, 0, -1,  0;
%e A226519   1, 2,  1, 2, 1, 0, -1,  0,  1, 0, -1, -2, -1, -2, -1, 0;
%e A226519   ...
%p A226519 with(numtheory);
%p A226519 T:=(n,k)->add(legendre(i,ithprime(n)),i=1..k);
%p A226519 f:=n->[seq(T(n,k),k=1..ithprime(n)-1)];
%p A226519 [seq(f(n),n=1..15)];
%t A226519 Table[P = Prime[n]; Table[JacobiSymbol[k,P], {k,P-1}]//Accumulate, {n,15}]// Flatten (* _G. C. Greubel_, Oct 05 2024 *)
%o A226519 (Magma)
%o A226519 A226519:= func< n,k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
%o A226519 [A226519(n,k) : k in [1..NthPrime(n)-1], n in [1..15]]; // _G. C. Greubel_, Oct 05 2024
%o A226519 (SageMath)
%o A226519 def A226519(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
%o A226519 flatten([[A226519(n,k) for k in range(1,nth_prime(n))] for n in range(1,16)]) # _G. C. Greubel_, Oct 05 2024
%Y A226519 A variant of A226518, which is the main entry for this triangle.
%Y A226519 Cf. A165582, A226518.
%K A226519 sign,tabf
%O A226519 1,9
%A A226519 _N. J. A. Sloane_, Jun 19 2013