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%I #7 Jun 13 2017 22:32:23
%S 1,-1,1,-3,-2,1,-15,-3,-3,1,-99,-15,-3,-4,1,-783,-99,-15,-3,-5,1,
%T -7083,-783,-99,-15,-3,-6,1,-71415,-7083,-783,-99,-15,-3,-7,1,-789939,
%U -71415,-7083,-783,-99,-15,-3,-8,1,-9485343,-789939,-71415,-7083,-783,-99,-15,-3,-9,1,-122721723,-9485343,-789939,-71415
%N Matrix inverse of triangle A111544.
%C The column sequences are all equal after the initial terms and are derived from the logarithm of a factorial series (cf. A111546).
%F T(n, n)=1 and T(n+1, n)=-n-1, else T(n+k+1, k) = -A111546(k) for k>=1.
%e Triangle begins:
%e 1;
%e -1,1;
%e -3,-2,1;
%e -15,-3,-3,1;
%e -99,-15,-3,-4,1;
%e -783,-99,-15,-3,-5,1;
%e -7083,-783,-99,-15,-3,-6,1;
%e -71415,-7083,-783,-99,-15,-3,-7,1;
%e -789939,-71415,-7083,-783,-99,-15,-3,-8,1; ...
%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-k,(i+2)!/2!*x^i)),n-k-1))))
%Y Cf. A111544, A111546, A111540.
%K sign,tabl
%O 0,4
%A _Paul D. Hanna_, Aug 07 2005
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