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A111978 Matrix log of triangle A111975, 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 A111978 #7 Jun 13 2017 22:38:49
%S A111978 0,1,0,0,2,0,16,0,4,0,0,32,0,8,0,1536,0,64,0,16,0,0,3072,0,128,0,32,0,
%T A111978 -319488,0,6144,0,256,0,64,0,0,-638976,0,12288,0,512,0,128,0,
%U A111978 36007575552,0,-1277952,0,24576,0,1024,0,256,0,0,72015151104,0,-2555904,0,49152,0,2048,0,512,0
%N A111978 Matrix log of triangle A111975, 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 A111978 Column k equals 2^k multiplied by column 0 (A111979) when ignoring zeros above the diagonal.
%F A111978 T(n, k) = 2^k*T(n-k, 0) = 2^k*A111979(n-k) for n>=k>=0.
%e A111978 Matrix log of A111975, with factorial denominators, begins:
%e A111978 0;
%e A111978 1/1!, 0;
%e A111978 0/2!, 2/1!, 0;
%e A111978 16/3!, 0/2!, 4/1!, 0;
%e A111978 0/4!, 32/3!, 0/2!, 8/1!, 0;
%e A111978 1536/5!, 0/4!, 64/3!, 0/2!, 16/1!, 0;
%e A111978 0/6!, 3072/5!, 0/4!, 128/3!, 0/2!, 32/1!, 0;
%e A111978 -319488/7!, 0/6!, 6144/5!, 0/4!, 256/3!, 0/2!, 64/1!, 0; ...
%o A111978 (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]=if(i>2,(A^q)[i-1,2],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 A111978 Cf. A111975 (triangle), A111979 (column 0).
%K A111978 frac,sign,tabl
%O A111978 0,5
%A A111978 _Paul D. Hanna_, Aug 25 2005