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%I #32 Oct 11 2021 18:43:27
%S 1,-1,2,0,-2,3,0,0,-3,4,0,0,0,-4,5,0,0,0,0,-5,6,0,0,0,0,0,-6,7,0,0,0,
%T 0,0,0,-7,8,0,0,0,0,0,0,0,-8,9,0,0,0,0,0,0,0,0,-9,10,0,0,0,0,0,0,0,0,
%U 0,-10,11,0,0,0,0,0,0,0,0,0,0,-11,12
%N Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0<k<=n.
%C The matrix inverse = (1/1; 1/2, 1/2; 1/3, 1/3, 1/3;...). Binomial transform of A128064 = A128065. A128064 * A007318 = A103406.
%C The positive version with row sums 2n+1 is given by T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(n-j)*(j+1)}. - _Paul Barry_, May 26 2007
%C Binomial transform of unsigned sequence is A003506. - _Gary W. Adamson_, Aug 29 2007
%C Table T(n,k) read by antidiagonals. T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - _Boris Putievskiy_, Feb 07 2013
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F Number triangle T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(j-k)*(j+1)}. - _Paul Barry_, May 26 2007
%F a(n) = A002260(n)*A167374(n); a(n) = i*floor((i+2)/(t+2))*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Feb 07 2013
%F G.f.: (-1)^k*[x^k*exp(k*x)]'/exp(k*x)=sum(n>=k, (-1)^n*T(n,k)*x^n). - _Vladimir Kruchinin_, Oct 18 2013
%e First few rows of the triangle are:
%e 1;
%e -1,2;
%e 0,-2,3;
%e 0,0,-3,4;
%e 0,0,0,-4,5;
%e 0,0,0,0,-5,6;
%e 0,0,0,0,0,-6,7;
%e ...
%e From _Boris Putievskiy_, Feb 07 2013: (Start)
%e The start of the sequence as table:
%e 1..-1..0..0..0..0..0...
%e 2..-2..0..0..0..0..0...
%e 3..-3..0..0..0..0..0...
%e 4..-4..0..0..0..0..0...
%e 5..-5..0..0..0..0..0...
%e 6..-6..0..0..0..0..0...
%e 7..-7..0..0..0..0..0...
%e . . .
%e (End)
%t row[1] = {1}; row[2] = {-1, 2}; row[n_] := Join[Array[0&, n-2], {-n+1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Jan 12 2015 *)
%Y Cf. A128065, A103406, A003506, A002260, A167374.
%K tabl,sign,easy
%O 1,3
%A _Gary W. Adamson_, Feb 14 2007
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