%I #23 Mar 22 2017 10:31:31
%S 1,1,1,-1,2,1,2,-1,3,1,-5,2,0,4,1,14,-5,1,2,5,1,-42,14,-3,0,5,6,1,132,
%T -42,9,-1,0,9,7,1,-429,132,-28,4,0,2,14,8,1,1430,-429,90,-14,1,0,7,20,
%U 9,1,-4862,1430,-297,48,-5,0,0,16,27,10,1
%N Triangle read by rows: T(n,m) = (-1)^(n-m-1)*m*binomial(2*n-3*m-1,n-m-1)/(n-m), T(n,n)=1.
%H Indranil Ghosh, <a href="/A271825/b271825.txt">Rows 1..100, flattened</a>
%F G.f.: -(x*sqrt(4*x+1)*y+x*y)/(x*sqrt(4*x+1)*y+x*y-2).
%F Sum_{n>=m) T(n,m)*x^n is expansion of (x*(1+sqrt(1+4*x))/2)^m.
%e Triangle begins
%e 1;
%e 1, 1;
%e -1, 2, 1;
%e 2, -1, 3, 1;
%e -5, 2, 0, 4, 1;
%e 14, -5, 1, 2, 5, 1;
%e -42, 14, -3, 0, 5, 6, 1;
%t Flatten[Table[If[n==m,1,(-1)^(n-m-1)*m*Binomial[2n-3m-1,n-m-1]/(n-m)],{n,1,11},{m,1,n}]] (* _Indranil Ghosh_, Feb 28 2017 *)
%o (Maxima)
%o taylor(1/(1-y*A(x))-1,x,0,10,y,0,10);
%o (PARI) a(n,m)=if(n==m, 1, (-1)^(n-m-1)*m*binomial(2*n-3*m-1,n-m-1)/(n-m));
%o tabl(nn)=for(n=1, nn, for(m=1, n, print1(a(n,m),", "));print)
%o tabl(11); \\ _Indranil Ghosh_, Feb 28 2017
%Y Cf. A000108.
%K sign,tabl
%O 1,5
%A _Vladimir Kruchinin_, Apr 14 2016