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A111810 Matrix log of triangle A098539, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.

%I A111810 #7 Jun 13 2017 22:33:21
%S A111810 0,1,0,2,2,0,10,4,4,0,88,20,8,8,0,1096,176,40,16,16,0,11856,2192,352,
%T A111810 80,32,32,0,-402480,23712,4384,704,160,64,64,0,-1891968,-804960,47424,
%U A111810 8768,1408,320,128,128,0,36024603264,-3783936,-1609920,94848,17536,2816,640,256,256,0
%N A111810 Matrix log of triangle A098539, which shifts columns left and up under matrix square; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%C A111810 Column k equals 2^k times column 0 (A111811) when ignoring zeros above the diagonal.
%F A111810 T(n, k) = 2^k*T(n-k, 0) = 2^k*A111811(n-k) for n>=k>=0.
%e A111810 Matrix log of A098539, with factorial denominators, begins:
%e A111810 0;
%e A111810 1/1!, 0;
%e A111810 2/2!, 2/1!, 0;
%e A111810 10/3!, 4/2!, 4/1!, 0;
%e A111810 88/4!, 20/3!, 8/2!, 8/1!, 0;
%e A111810 1096/5!, 176/4!, 40/3!, 16/2!, 16/1!, 0;
%e A111810 11856/6!, 2192/5!, 352/4!, 80/3!, 32/2!, 32/1!, 0; ...
%o A111810 (PARI) T(n,k,q=2)=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 A111810 Cf. A098539 (triangle), A111811 (column 0), A111813 (variant), A111941 (q=-1), A111843 (q=3), A111848 (q=4).
%K A111810 frac,sign,tabl
%O A111810 0,4
%A A111810 _Paul D. Hanna_, Aug 22 2005