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A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111815 #8 Jun 13 2017 22:33:49
%S A111815 0,1,0,-1,3,0,-3,-3,9,0,150,-9,-9,27,0,1236,450,-27,-27,81,0,-2555748,
%T A111815 3708,1350,-81,-81,243,0,-64342116,-7667244,11124,4050,-243,-243,729,
%U A111815 0,5885700899760,-193026348,-23001732,33372,12150,-729,-729,2187,0
%N A111815 Matrix log of triangle A078122, which shifts columns left and up under matrix cube; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111815 Column k equals 3^k multiplied by column 0 (A111816) when ignoring zeros above the diagonal.
%F A111815 T(n, k) = 3^k*T(n-k, 0) = A111816(n-k) for n>=k>=0.
%e A111815 Matrix log of A078122, with factorial denominators, begins:
%e A111815 0;
%e A111815 1/1!, 0;
%e A111815 -1/2!, 3/1!, 0;
%e A111815 -3/3!, -3/2!, 9/1!, 0;
%e A111815 150/4!, -9/3!, -9/2!, 27/1!, 0;
%e A111815 1236/5!, 450/4!, -27/3!, -27/2!, 81/1!, 0;
%e A111815 -2555748/6!, 3708/5!, 1350/4!, -81/3!, -81/2!, 243/1!, 0; ...
%o A111815 (PARI) T(n,k,q=3)=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 A111815 Cf. A078122, A111816 (column 0), A111840 (variant); log matrices: A110504 (q=-1), A111813 (q=2), A111818 (q=4), A111823 (q=5), A111828 (q=6), A111833 (q=7), A111838 (q=8).
%K A111815 frac,sign,tabl
%O A111815 0,5
%A A111815 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005