|
%I #21 Dec 09 2015 08:43:03
%S 0,1,0,-1,-1,0,1,1,1,0,-2,-1,-1,-1,0,4,2,1,1,1,0,-12,-4,-2,-1,-1,-1,0,
%T 36,12,4,2,1,1,1,0,-144,-36,-12,-4,-2,-1,-1,-1,0,576,144,36,12,4,2,1,
%U 1,1,0,-2880,-576,-144,-36,-12,-4,-2,-1,-1,-1,0,14400,2880,576,144,36,12,4,2,1,1,1,0,-86400,-14400,-2880,-576,-144,-36
%N Matrix log of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying each element in row n and column k by (n-k)!.
%F T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0.
%e Triangle begins:
%e 0;
%e 1, 0;
%e -1, -1, 0;
%e 1, 1, 1, 0;
%e -2, -1, -1, -1, 0;
%e 4, 2, 1, 1, 1, 0;
%e -12, -4, -2, -1, -1, -1, 0;
%e 36, 12, 4, 2, 1, 1, 1, 0;
%e -144, -36, -12, -4, -2, -1, -1, -1, 0;
%e 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
%e -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
%e 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
%e -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0;
%e 518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0;
%e -3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ...
%e where, apart from signs, the columns are all the same (A111942).
%e ...
%e Triangle A111940 begins:
%e 1;
%e 1, 1;
%e -1, -1, 1;
%e 0, 0, 1, 1;
%e 0, 0, -1, -1, 1;
%e 0, 0, 0, 0, 1, 1;
%e 0, 0, 0, 0, -1, -1, 1;
%e 0, 0, 0, 0, 0, 0, 1 ,1;
%e 0, 0, 0, 0, 0, 0, -1, -1, 1; ...
%e where the matrix inverse shifts columns left and up one place.
%e ...
%e The matrix log of A111940, with factorial denominators, begins:
%e 0;
%e 1/1!, 0;
%e -1/2!, -1/1!, 0;
%e 1/3!, 1/2!, 1/1!, 0;
%e -2/4!, -1/3!, -1/2!, -1/1!, 0;
%e 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
%e -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
%e 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
%e -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
%e 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0;
%e -2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0;
%e 14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ...
%e Note that the square of the matrix log of A111940 begins:
%e 0;
%e 0, 0;
%e -1, 0, 0;
%e 0, -1, 0, 0;
%e -1/12, 0, -1, 0, 0;
%e 0, -1/12, 0, -1, 0, 0;
%e -1/90, 0, -1/12, 0, -1, 0, 0;
%e 0, -1/90, 0, -1/12, 0, -1, 0, 0;
%e -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
%e 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
%e -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
%e 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0;
%e -1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ...
%e where nonzero terms are negative unit fractions with denominators given by A002544:
%e [1, 12, 90, 560, 3150, 16632, 84084, 411840, ..., C(2*n+1,n)*(n+1)^2, ...].
%o (PARI) {T(n,k,q=-1) = 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]))}
%o for(n=0, 16, for(k=0, n, print1(T(n, k, -1), ", ")); print(""))
%Y Cf. A111940 (triangle), A111942 (column 0), A110504 (variant).
%K frac,sign,tabl
%O 0,11
%A _Paul D. Hanna_, Aug 23 2005
|
