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A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111823 #10 Jun 13 2017 22:34:16
%S A111823 0,1,0,-3,5,0,16,-15,25,0,2814,80,-75,125,0,-1092180,14070,400,-375,
%T A111823 625,0,-3603928080,-5460900,70350,2000,-1875,3125,0,58978973128440,
%U A111823 -18019640400,-27304500,351750,10000,-9375,15625,0,5974833380453777520
%N A111823 Matrix log of triangle A111820, which shifts columns left and up under matrix 5th power; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111823 Column k equals 5^k multiplied by column 0 (A111824) when ignoring zeros above the diagonal.
%F A111823 T(n, k) = 5^k*T(n-k, 0) = A111824(n-k) for n>=k>=0.
%e A111823 Matrix log of A111820, with factorial denominators, begins:
%e A111823 0;
%e A111823 1/1!, 0;
%e A111823 -3/2!, 5/1!, 0;
%e A111823 16/3!, -15/2!, 25/1!, 0;
%e A111823 2814/4!, 80/3!, -75/2!, 125/1!, 0;
%e A111823 -1092180/5!, 14070/4!, 400/3!, -375/2!, 625/1!, 0; ...
%o A111823 (PARI) T(n,k,q=5)=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 A111823 Cf. A111820, A111824 (column 0); log matrices: A110504 (q=-1), A111813 (q=2), A111815 (q=3), A111818 (q=4), A111828 (q=6), A111833 (q=7), A111838 (q=8).
%K A111823 frac,sign,tabl
%O A111823 0,4
%A A111823 _Gottfried Helms_ and _Paul D. Hanna_, Aug 22 2005