OFFSET
0,4
COMMENTS
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x+1)^1 + A_2*(x-1)^2 + A_3*(x+1)^3 + ... + A_n*(x-(-1)^n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
FORMULA
T(n,n-1) = 1 + n*(-1)^n for n > 0.
T(n,n-2) = (1-n)*((3/2)*n+(-1)^n) + 1 for n > 1.
Rows sum to 1.
EXAMPLE
1;
0, 1;
-3, 3, 1;
8, -6, -2, 1;
65, -50, -20, 5, 1;
-296, 235, 90, -25, -4, 1;
-3059, 2401, 945, -245, -49, 7, 1;
21552, -16940, -6636, 1750, 336, -56, -6, 1;
272289, -213828, -84000, 21966, 4326, -672, -90, 9, 1;
-2600752, 2042805, 801996, -210126, -41160, 6570, 834, -99, -8, 1
PROG
(PARI) a(n, j)=if(j==n, return(1)); if(j!=n, return(1-sum(i=1, n-j, (-1)^(i*j)*binomial(i+j, i)*a(n, i+j))))
for(n=0, 15, for(j=0, n, print1(a(n, j), ", ")))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Derek Orr, Oct 18 2014
STATUS
approved