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A111813 Matrix log of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111813 #8 Jun 13 2017 22:33:29
%S A111813 0,1,0,0,2,0,-2,0,4,0,0,-4,0,8,0,216,0,-8,0,16,0,0,432,0,-16,0,32,0,
%T A111813 -568464,0,864,0,-32,0,64,0,0,-1136928,0,1728,0,-64,0,128,0,
%U A111813 36058658688,0,-2273856,0,3456,0,-128,0,256,0,0,72117317376,0,-4547712,0,6912,0,-256,0,512,0
%N A111813 Matrix log of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111813 Column k equals 2^k multiplied by column 0 (A111814) when ignoring zeros above the diagonal.
%F A111813 T(n, k) = 2^k*T(n-k, 0) = A111814(n-k) for n>=k>=0.
%e A111813 Matrix log of A078121, with factorial denominators, begins:
%e A111813 0;
%e A111813 1/1!, 0;
%e A111813 0/2!, 2/1!, 0;
%e A111813 -2/3!, 0/2!, 4/1!, 0;
%e A111813 0/4!, -4/3!, 0/2!, 8/1!, 0;
%e A111813 216/5!, 0/4!, -8/3!, 0/2!, 16/1!, 0;
%e A111813 0/6!, 432/5!, 0/4!, -16/3!, 0/2!, 32/1!, 0;
%e A111813 -568464/7!, 0/6!, 864/5!, 0/4!, -32/3!, 0/2!, 64/1!, 0; ...
%o A111813 (PARI) T(n,k,q=2)=local(A=Mat(1),B);if(n<k || k<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); B=sum(i=1,#A,-(A^0-A)^i/i);return((n-k)!*B[n+1,k+1]))
%Y A111813 Cf. A078121, A111814 (column 0), A111810 (variant); log matrices: A110504 (q=-1), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).
%K A111813 frac,sign,tabl
%O A111813 0,5
%A A111813 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005