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Triangle, read by rows, that equals the matrix inverse of A071207 when treated as a lower triangular matrix.
1

%I #12 Jan 05 2021 21:49:08

%S 1,-1,1,0,-4,1,0,9,-9,1,0,-16,48,-16,1,0,25,-200,150,-25,1,0,-36,720,

%T -1080,360,-36,1,0,49,-2352,6615,-3920,735,-49,1,0,-64,7168,-36288,

%U 35840,-11200,1344,-64,1,0,81,-20736,183708,-290304,141750,-27216,2268,-81,1,0,-100,57600,-874800,2150400,-1575000,453600

%N Triangle, read by rows, that equals the matrix inverse of A071207 when treated as a lower triangular matrix.

%C A071207 describes the transform of a sequence B that results in a sequence D defined by: "d(n) = the (n+1)-th term of the n-th binomial transform of sequence B". Then d(n) = Sum_{k=0..n} A071207(n,k)*b(k) for n >= 0, where A071207(n,k) = n^(n-k)*C(n,k). The matrix inverse of A071207 describes the inverse transform that yields B from D: b(n) = Sum_{k=0..n} T(n,k)*d(k) for n >= 0, where T(0,0)=1, T(n,k) = (-1)^(n-k)*C(n,k)*n*k^(n-k-1) for 0 < k <= n.

%F T(n, k) = (-1)^(n-k)*C(n, k)*n*k^(n-k-1) for 0 < k <= n, with T(0, 0)=1.

%F O.g.f.: A(x, y) = (1-y)*Sum_{n>=0} x^n*y^n/(1+n*y)^(n+2).

%F E.g.f.: (1-y)*exp(x*y*exp(-y)). - _Vladeta Jovovic_, Nov 18 2003

%e Rows begin:

%e 1;

%e -1, 1;

%e 0, -4, 1;

%e 0, 9, -9, 1;

%e 0, -16, 48, -16, 1;

%e 0, 25, -200, 150, -25, 1;

%e 0, -36, 720, -1080, 360, -36, 1;

%e 0, 49, -2352, 6615, -3920, 735, -49, 1;

%e 0, -64, 7168, -36288, 35840, -11200, 1344, -64, 1;

%o (PARI) T(n,k)=if(k<0 || k>n,0,if(n==0 && k==0,1,(-1)^(n-k)*binomial(n,k)*n*k^(n-k-1)))

%Y Cf. A071207.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 17 2003