cp's OEIS Frontend

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

%I A104416 #9 Jan 18 2025 07:43:31
%S A104416 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 A104416 1,0,0,0,24,-225,175,-28,1,0,0,0,0,274,-735,322,-36,1,0,0,0,0,-120,
%U A104416 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 A104416 Triangle, read by rows, where T(n,k) = A008275(k+1,n-k+1) are Stirling numbers of the first kind.
%C A104416 The matrix inverse forms A104417, in which column 0 equals A082161.
%F A104416 G.f.: A(x, y) = Sum_{n>=0} x^n*y^n*Product_{k=1..n+1} (1-k*x).
%e A104416 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) + ...
%e A104416 Rows begin:
%e A104416   1;
%e A104416   -1,1;
%e A104416   0,-3,1;
%e A104416   0,2,-6,1;
%e A104416   0,0,11,-10,1;
%e A104416   0,0,-6,35,-15,1;
%e A104416   0,0,0,-50,85,-21,1;
%e A104416   0,0,0,24,-225,175,-28,1;
%e A104416   0,0,0,0,274,-735,322,-36,1;
%e A104416   0,0,0,0,-120,1624,-1960,546,-45,1;
%e A104416   ...
%o A104416 (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 A104416 Cf. A104417, A082161, A008275.
%K A104416 sign,tabl
%O A104416 0,5
%A A104416 _Paul D. Hanna_, Mar 06 2005