%I #6 Feb 03 2025 21:35:01

%S 1,-1,1,2,-3,1,-9,15,-7,1,94,-160,80,-15,1,-2220,3790,-1915,375,-31,1,

%T 114456,-195461,98875,-19460,1652,-63,1,-12542341,21419587,-10836231,

%U 2133635,-181559,7035,-127,1,2868686486,-4899099640,2478483560,-488022556,41534164,-1611120,29360,-255,1

%N Matrix inverse of A008278, which is the reflected triangle of the Stirling numbers of 2nd kind.

%F T(n, k) = (Stirling2(n, n-k))^[-1], where T^[-1] denotes the matrix inverse of T.

%e Triangle T begins:

%e 1;

%e -1, 1;

%e 2, -3, 1;

%e -9, 15, -7, 1;

%e 94, -160, 80, -15, 1;

%e -2220, 3790, -1915, 375, -31, 1;

%e 114456, -195461, 98875, -19460, 1652, -63, 1;

%e -12542341, 21419587, -10836231, 2133635, -181559, 7035, -127, 1;

%o (PARI) {T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c, sum(m=0,r-c+1,(-1)^(r-c+1-m)*m^r/m!/(r-c+1-m)!)))^-1)[n+1,k+1]}

%Y Row sums are A000007.

%Y Column 0 is A106343.

%Y Cf. A008278, A106340.

%K sign,tabl

%O 0,4

%A _Paul D. Hanna_, May 01 2005