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A106340 Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = (n-k)!*A008278(n+1,k+1), for n>=k>=0, where A008278 is a triangle of Stirling numbers of 2nd kind.

%I A106340 #12 Jun 27 2019 06:11:12
%S A106340 1,-1,1,1,-3,1,-1,9,-7,1,1,-45,55,-15,1,-1,585,-835,285,-31,1,1,
%T A106340 -21105,30835,-11025,1351,-63,1,-1,1858185,-2719675,977445,-121891,
%U A106340 6069,-127,1,1,-367958745,538607755,-193649085,24187051,-1213065,26335,-255,1,-1,157169540745,-230061795355,82717588485
%N A106340 Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = (n-k)!*A008278(n+1,k+1), for n>=k>=0, where A008278 is a triangle of Stirling numbers of 2nd kind.
%C A106340 Row sums are {1,0,-1,2,-3,4,-5,6,...}. Column 1 is A106341.
%F A106340 T(n, k) = A106338(n, k)/k!, for n>=k>=0.
%e A106340 Triangle T begins:
%e A106340   1;
%e A106340   -1,1;
%e A106340   1,-3,1;
%e A106340   -1,9,-7,1;
%e A106340   1,-45,55,-15,1;
%e A106340   -1,585,-835,285,-31,1;
%e A106340   1,-21105,30835,-11025,1351,-63,1;
%e A106340   -1,1858185,-2719675,977445,-121891,6069,-127,1;
%e A106340   1,-367958745,538607755,-193649085,24187051,-1213065,26335,-255,1;
%e A106340   ...
%e A106340 Matrix inverse begins:
%e A106340   1;
%e A106340   1,1;
%e A106340   2,3,1;
%e A106340   6,12,7,1;
%e A106340   24,60,50,15,1;
%e A106340   120,360,390,180,31,1;
%e A106340   ...
%e A106340 where [T^-1](n,k) = (n-k)!*A008278(n+1,k+1).
%t A106340 rows = 10;
%t A106340 M = Table[If[r >= c, (r-c)! Sum[(-1)^(r-c-m+1) m^r/m!/(r-c-m+1)!, {m, 0, r-c+1}], 0], {r, rows}, {c, rows}] // Inverse;
%t A106340 T[n_, k_] := M[[n+1, k+1]];
%t A106340 Table[T[n, k], {n, 0, rows-1}, {k, 0, n}] (* _Jean-François Alcover_, Jun 27 2019, from PARI *)
%o A106340 (PARI) {T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c,(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]}
%o A106340 (Sage)
%o A106340 def A106340_matrix(d):
%o A106340     def A130850(n, k):   # EulerianNumber = A173018
%o A106340         return add(EulerianNumber(n,j)*binomial(n-j,k) for j in (0..n))
%o A106340     return matrix(ZZ, d, A130850).inverse()
%o A106340 A106340_matrix(8)  # _Peter Luschny_, May 21 2013
%Y A106340 Cf. A106338, A008278, A106341.
%K A106340 sign,tabl
%O A106340 0,5
%A A106340 _Paul D. Hanna_, May 01 2005