A271825 - OEIS

%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