A104416 - OEIS

%I #4 Mar 30 2012 18:36:45

%S 1,-1,1,0,-3,1,0,2,-6,1,0,0,11,-10,1,0,0,-6,35,-15,1,0,0,0,-50,85,-21,

%T 1,0,0,0,24,-225,175,-28,1,0,0,0,0,274,-735,322,-36,1,0,0,0,0,-120,

%U 1624,-1960,546,-45,1,0,0,0,0,0,-1764,6769,-4536,870,-55,1,0,0,0,0,0,720,-13132,22449,-9450,1320,-66,1

%N Triangle, read by rows, where T(n,k) = A008275(k+1,n-k+1) are Stirling numbers of the first kind.

%C The matrix inverse forms A104417, in which column 0 equals A082161.

%F G.f.: A(x, y) = Sum_{n>=0} x^n*y^n*Product(k=1..n+1} (1-k*x).

%e A(x,y) = (1-x) + x*y*(1-x)*(1-2*x) + x^2*y^2*(1-x)*(1-2*x)*(1-3*x) +

%e x^3*y^3*(1-x)*(1-2*x)*(1-3*x)*(1-4*x) + ...

%e Rows begin:

%e 1;

%e -1,1;

%e 0,-3,1;

%e 0,2,-6,1;

%e 0,0,11,-10,1;

%e 0,0,-6,35,-15,1;

%e 0,0,0,-50,85,-21,1;

%e 0,0,0,24,-225,175,-28,1;

%e 0,0,0,0,274,-735,322,-36,1;

%e 0,0,0,0,-120,1624,-1960,546,-45,1; ...

%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff(sum(i=0,n,X^i*Y^i*prod(j=1,i+1,1-j*X)),n,x),k,y)}

%Y Cf. A104417, A082161.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 06 2005