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A111848 Matrix log of triangle A111845, 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 A111848 #9 Jun 13 2017 22:38:32
%S A111848 0,1,0,4,4,0,56,16,16,0,1728,224,64,64,0,-45696,6912,896,256,256,0,
%T A111848 -159401472,-182784,27648,3584,1024,1024,0,387212983296,-637605888,
%U A111848 -731136,110592,14336,4096,4096,0,14722642769657856,1548851933184,-2550423552,-2924544,442368,57344,16384,16384,0
%N A111848 Matrix log of triangle A111845, 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 A111848 Column k equals 4^k multiplied by column 0 (A111849) when ignoring zeros above the diagonal.
%F A111848 T(n, k) = 4^k*T(n-k, 0) = 4^k*A111844(n-k) for n>=k>=0.
%e A111848 Matrix log of A111845, with factorial denominators, begins:
%e A111848 0;
%e A111848 1/1!, 0;
%e A111848 4/2!, 4/1!, 0;
%e A111848 56/3!, 16/2!, 16/1!, 0;
%e A111848 1728/4!, 224/3!, 64/2!, 64/1!, 0;
%e A111848 -45696/5!, 6912/4!, 896/3!, 256/2!, 256/1!, 0; ...
%o A111848 (PARI) L(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,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], 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 A111848 Cf. A111845 (triangle), A111849 (column 0), A111818 (variant).
%K A111848 frac,sign,tabl
%O A111848 0,4
%A A111848 _Paul D. Hanna_, Aug 23 2005