OFFSET
0,4
COMMENTS
Row sums are all zeros after row 0.
FORMULA
T(n,k) = Sum_{i=k..n} C(n,i)*T(i-1,k-1) for k>0, with T(0,0)=1 and T(n,0)=-n for n>0.
EXAMPLE
To generate, start with [1,-1,-2,-3,-4,...] in column 0.
Then column 1 = BINOMIAL[0, 1,-1,-2,-3,-4,-5,...]
= [0,1,1,-2,-13,-44,-123,-314,-761,...];
column 2 = BINOMIAL[0, 0,1,1,-2,-13,-44,-123,-314,...]
= [0,0,1,4,8,-3,-117,-718,-3314,-13481,...];
column 3 = BINOMIAL[0, 0,0,1,4,8,-3,-117,-718,...]
= [0,0,0,1,8,38,125,205,-954,-12644,-90625,...]; etc.
Triangle begins:
1;
-1,1;
-2,1,1;
-3,-2,4,1;
-4,-13,8,8,1;
-5,-44,-3,38,13,1;
-6,-123,-117,125,101,19,1;
-7,-314,-718,205,594,213,26,1;
-8,-761,-3314,-954,2787,1822,393,34,1;
-9,-1784,-13481,-12644,9717,12987,4507,663,43,1;
-10,-4087,-51055,-90625,12247,79419,43282,9727,1048,53,1; ...
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(k==0, -n, sum(i=k, n, binomial(n, i)*T(i-1, k-1)))))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Mar 08 2006
STATUS
approved