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%I #9 Jun 13 2017 23:41:10
%S 1,-1,1,-2,1,1,-3,-2,4,1,-4,-13,8,8,1,-5,-44,-3,38,13,1,-6,-123,-117,
%T 125,101,19,1,-7,-314,-718,205,594,213,26,1,-8,-761,-3314,-954,2787,
%U 1822,393,34,1,-9,-1784,-13481,-12644,9717,12987,4507,663,43,1,-10,-4087,-51055,-90625,12247,79419,43282,9727,1048
%N Triangle, read by rows, where column 0 is [1,-1,-2,-3,...,-n,...] and column k+1 is generated by the binomial transform of column k preceded by a zero (column k includes the k zeros above the main diagonal).
%C Row sums are all zeros after row 0.
%F 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.
%e To generate, start with [1,-1,-2,-3,-4,...] in column 0.
%e Then column 1 = BINOMIAL[0, 1,-1,-2,-3,-4,-5,...]
%e = [0,1,1,-2,-13,-44,-123,-314,-761,...];
%e column 2 = BINOMIAL[0, 0,1,1,-2,-13,-44,-123,-314,...]
%e = [0,0,1,4,8,-3,-117,-718,-3314,-13481,...];
%e column 3 = BINOMIAL[0, 0,0,1,4,8,-3,-117,-718,...]
%e = [0,0,0,1,8,38,125,205,-954,-12644,-90625,...]; etc.
%e Triangle begins:
%e 1;
%e -1,1;
%e -2,1,1;
%e -3,-2,4,1;
%e -4,-13,8,8,1;
%e -5,-44,-3,38,13,1;
%e -6,-123,-117,125,101,19,1;
%e -7,-314,-718,205,594,213,26,1;
%e -8,-761,-3314,-954,2787,1822,393,34,1;
%e -9,-1784,-13481,-12644,9717,12987,4507,663,43,1;
%e -10,-4087,-51055,-90625,12247,79419,43282,9727,1048,53,1; ...
%o (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)))))
%Y Cf. A117335 (matrix inverse), A117336, A117337, A117338.
%K sign,tabl
%O 0,4
%A _Paul D. Hanna_, Mar 08 2006
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