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A111833 Matrix log of triangle A111830, which shifts columns left and up under matrix 7th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111833 #10 Jun 13 2017 22:36:41
%S A111833 0,1,0,-5,7,0,83,-35,49,0,16110,581,-245,343,0,-40097784,112770,4067,
%T A111833 -1715,2401,0,-388036363380,-280684488,789390,28469,-12005,16807,0,
%U A111833 82804198261002036,-2716254543660,-1964791416,5525730,199283,-84035,117649,0
%N A111833 Matrix log of triangle A111830, which shifts columns left and up under matrix 7th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111833 Column k equals 7^k multiplied by column 0 (A111834) when ignoring zeros above the diagonal.
%F A111833 T(n, k) = 7^k*T(n-k, 0) = A111834(n-k) for n>=k>=0.
%e A111833 Matrix log of A111830, with factorial denominators, begins:
%e A111833 0;
%e A111833 1/1!, 0;
%e A111833 -5/2!, 7/1!, 0;
%e A111833 83/3!, -35/2!, 49/1!, 0;
%e A111833 16110/4!, 581/3!, -245/2!, 343/1!, 0;
%e A111833 -40097784/5!, 112770/4!, 4067/3!, -1715/2!, 2401/1!, 0; ...
%o A111833 (PARI) T(n,k,q=7)=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 A111833 Cf. A111830, A111834 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111838 (q=8).
%K A111833 frac,sign,tabl
%O A111833 0,4
%A A111833 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005