OFFSET
0,5
COMMENTS
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.
FORMULA
T(n, k) = (-1)^(n-k)*C(n, k)*n*k^(n-k-1) for 0 < k <= n, with T(0, 0)=1.
O.g.f.: A(x, y) = (1-y)*Sum_{n>=0} x^n*y^n/(1+n*y)^(n+2).
E.g.f.: (1-y)*exp(x*y*exp(-y)). - Vladeta Jovovic, Nov 18 2003
EXAMPLE
Rows begin:
1;
-1, 1;
0, -4, 1;
0, 9, -9, 1;
0, -16, 48, -16, 1;
0, 25, -200, 150, -25, 1;
0, -36, 720, -1080, 360, -36, 1;
0, 49, -2352, 6615, -3920, 735, -49, 1;
0, -64, 7168, -36288, 35840, -11200, 1344, -64, 1;
PROG
(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)))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Nov 17 2003
STATUS
approved