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A106342 Matrix inverse of A008278, which is the reflected triangle of the Stirling numbers of 2nd kind.

%I A106342 #6 Feb 03 2025 21:35:01
%S A106342 1,-1,1,2,-3,1,-9,15,-7,1,94,-160,80,-15,1,-2220,3790,-1915,375,-31,1,
%T A106342 114456,-195461,98875,-19460,1652,-63,1,-12542341,21419587,-10836231,
%U A106342 2133635,-181559,7035,-127,1,2868686486,-4899099640,2478483560,-488022556,41534164,-1611120,29360,-255,1
%N A106342 Matrix inverse of A008278, which is the reflected triangle of the Stirling numbers of 2nd kind.
%F A106342 T(n, k) = (Stirling2(n, n-k))^[-1], where T^[-1] denotes the matrix inverse of T.
%e A106342 Triangle T begins:
%e A106342           1;
%e A106342          -1,        1;
%e A106342           2,       -3,         1;
%e A106342          -9,       15,        -7,       1;
%e A106342          94,     -160,        80,     -15,       1;
%e A106342       -2220,     3790,     -1915,     375,     -31,    1;
%e A106342      114456,  -195461,     98875,  -19460,    1652,  -63,    1;
%e A106342   -12542341, 21419587, -10836231, 2133635, -181559, 7035, -127, 1;
%o A106342 (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 A106342 Row sums are A000007.
%Y A106342 Column 0 is A106343.
%Y A106342 Cf. A008278, A106340.
%K A106342 sign,tabl
%O A106342 0,4
%A A106342 _Paul D. Hanna_, May 01 2005