A104416 Triangle, read by rows, where T(n,k) = A008275(k+1,n-k+1) are Stirling numbers of the first kind.

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

Offset: 0

Author

Paul D. Hanna, Mar 06 2005

Keywords

• sign
• tabl

Comments

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

Examples

			A(x,y) = (1-x) + x*y*(1-x)*(1-2*x) + x^2*y^2*(1-x)*(1-2*x)*(1-3*x) + x^3*y^3*(1-x)*(1-2*x)*(1-3*x)*(1-4*x) + ...
Rows begin:
  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,-21,1;
  0,0,0,24,-225,175,-28,1;
  0,0,0,0,274,-735,322,-36,1;
  0,0,0,0,-120,1624,-1960,546,-45,1;
  ...
		

Crossrefs

Cf. A104417, A082161, A008275.

Programs

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)}

Formula

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