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 A111843 Matrix log of triangle A111840, 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)!. 4

%I

%S 0,1,0,3,3,0,27,9,9,0,486,81,27,27,0,7776,1458,243,81,81,0,-2423196,

%T 23328,4374,729,243,243,0,-97338996,-7269588,69984,13122,2187,729,729,

%U 0,5883879500784,-292016988,-21808764,209952,39366,6561,2187,2187,0

%N Matrix log of triangle A111840, 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 Column k equals 3^k multiplied by column 0 (A111844) when ignoring zeros above the diagonal.

%F T(n, k) = 3^k*T(n-k, 0) = 3^k*A111844(n-k) for n>=k>=0.

%e Matrix log of A111840, with factorial denominators, begins:

%e 0;

%e 1/1!, 0;

%e 3/2!, 3/1!, 0;

%e 27/3!, 9/2!, 9/1!, 0;

%e 486/4!, 81/3!, 27/2!, 27/1!, 0;

%e 7776/5!, 1458/4!, 243/3!, 81/2!, 81/1!, 0;

%e -2423196/6!, 23328/5!, 4374/4!, 729/3!, 243/2!, 243/1!, 0;

%o (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,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 Cf. A111840 (triangle), A111844 (column 0), A111815 (variant), A111941 (q=-1), A111810 (q=2), A111848 (q=4).

%K frac,sign,tabl

%O 0,4

%A _Paul D. Hanna_, Aug 23 2005

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