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%I #29 Nov 29 2023 06:58:33
%S 1,0,1,0,-1,1,0,0,-1,1,0,0,1,-1,1,0,0,0,1,-1,1,0,0,0,-1,1,-1,1,0,0,0,
%T 0,-1,1,-1,1,0,0,0,0,1,-1,1,-1,1,0,0,0,0,0,1,-1,1,-1,1,0,0,0,0,0,-1,1,
%U -1,1,-1,1,0,0,0,0,0,0,-1,1,-1,1,-1,1,0,0,0,0,0,0,1,-1,1
%N Number triangle T(n,k) = (-1)^(n-k) if binomial(k, n-k) > 0, 0 otherwise, with 0 <= k <= n.
%C Alternating sign version of A101688. A187036 is an eigensequence. Diagonal sums are A187035. Row sums are A133872.
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations (of) Integer Sequences And Pairing Functions</a>, 2012, arXiv:1212.2732 [math.CO], 2012.
%F From _Boris Putievskiy_, Jan 09 2013: (Start)
%F a(n) = A101688(n)*(-1)^(A003056(n) + A002260(n) + 1).
%F a(n) = floor((2*A002260(n)+1)/(A003056(n)+3))*(-1)^(A003056(n) + A002260(n) + 1).
%F a(n) = floor((2*n-t*(t+1)+1)/(t+3))*(-1)^(n-t*(t-1)/2+1), n > 0, where t = floor((-1+sqrt(8*n-7))/2). (End)
%e Triangle begins
%e 1;
%e 0, 1;
%e 0, -1, 1;
%e 0, 0, -1, 1;
%e 0, 0, 1, -1, 1;
%e 0, 0, 0, 1, -1, 1;
%e 0, 0, 0, -1, 1, -1, 1;
%e 0, 0, 0, 0, -1, 1, -1, 1;
%e 0, 0, 0, 0, 1, -1, 1, -1, 1;
%e 0, 0, 0, 0, 0, 1, -1, 1, -1, 1;
%e 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, 1;
%e 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1;
%t T[n_, k_] := Boole[n <= 2k] (-1)^(n-k);
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 05 2018 *)
%o (PARI) T(n,k)=if(n<=2*k,(-1)^(n-k),0) \\ _Charles R Greathouse IV_, Dec 28 2011
%K sign,tabl,easy
%O 0
%A _Paul Barry_, Mar 08 2011
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