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%I #6 Jun 13 2017 22:31:43
%S 1,-1,1,-2,-2,1,-8,-2,-3,1,-44,-8,-2,-4,1,-296,-44,-8,-2,-5,1,-2312,
%T -296,-44,-8,-2,-6,1,-20384,-2312,-296,-44,-8,-2,-7,1,-199376,-20384,
%U -2312,-296,-44,-8,-2,-8,1,-2138336,-199376,-20384,-2312,-296,-44,-8,-2,-9,1,-24936416,-2138336,-199376,-20384,-2312,-296
%N Matrix inverse of triangle A111536.
%C The column sequences are derived from the logarithm of a factorial series (cf. A111537).
%F T(n, n)=1 and T(n+1, n)=n+1, else T(n+k+1, k) = -A111537(k) for k>=1.
%e Triangle begins:
%e 1;
%e -1,1;
%e -2,-2,1;
%e -8,-2,-3,1;
%e -44,-8,-2,-4,1;
%e -296,-44,-8,-2,-5,1;
%e -2312,-296,-44,-8,-2,-6,1;
%e -20384,-2312,-296,-44,-8,-2,-7,1;
%e -199376,-20384,-2312,-296,-44,-8,-2,-8,1; ...
%e After initial terms, all columns are equal to -A111537.
%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,-n, -(n-k-1)*polcoeff(log(sum(i=0,n,(i+1)!/1!*x^i)),n-k-1))))
%Y Cf. A111536, A111537.
%K sign,tabl
%O 0,4
%A _Paul D. Hanna_, Aug 06 2005
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