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A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111818 #10 Jun 13 2017 22:34:23
%S A111818 0,1,0,-2,4,0,2,-8,16,0,840,8,-32,64,0,-76056,3360,32,-128,256,0,
%T A111818 -158761104,-304224,13440,128,-512,1024,0,390564896784,-635044416,
%U A111818 -1216896,53760,512,-2048,4096,0,14713376473366656,1562259587136,-2540177664,-4867584,215040,2048,-8192,16384,0
%N A111818 Matrix log of triangle A078536, which shifts columns left and up under matrix 4th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111818 Column k equals 4^k multiplied by column 0 (A111819) when ignoring zeros above the diagonal.
%F A111818 T(n, k) = 4^k*T(n-k, 0) = A111819(n-k) for n>=k>=0.
%e A111818 Matrix log of A078536, with factorial denominators, begins:
%e A111818 0;
%e A111818 1/1!, 0;
%e A111818 -2/2!, 4/1!, 0;
%e A111818 2/3!, -8/2!, 16/1!, 0;
%e A111818 840/4!, 8/3!, -32/2!, 64/1!, 0;
%e A111818 -76056/5!, 3360/4!, 32/3!, -128/2!, 256/1!, 0;
%e A111818 -158761104/6!, -304224/5!, 13440/4!, 128/3!, -512/2!, 1024/1!, 0;
%o A111818 (PARI) T(n,k,q=4)=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 A111818 Cf. A078536, A111819 (column 0), A111845 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).
%K A111818 frac,sign,tabl
%O A111818 0,4
%A A111818 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005