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 A111941 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)!. 6
 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, 36, 12, 4, 2, 1, 1, 1, 0, -144, -36, -12, -4, -2, -1, -1, -1, 0, 576, 144, 36, 12, 4, 2, 1, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 LINKS FORMULA T(n, k) = (-1)^k*T(n-k, 0) = (-1)^k*A111942(n-k) for n>=k>=0. EXAMPLE Triangle begins: 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; 36, 12, 4, 2, 1, 1, 1, 0; -144, -36, -12, -4, -2, -1, -1, -1, 0; 576, 144, 36, 12, 4, 2, 1, 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, -12, -4, -2, -1, -1, -1, 0; 518400, 86400, 14400, 2880, 576, 144, 36, 12, 4, 2, 1, 1, 1, 0; -3628800, -518400, -86400, -14400, -2880, -576, -144, -36, -12, -4, -2, -1, -1, -1, 0; ... where, apart from signs, the columns are all the same (A111942). ... Triangle A111940 begins: 1; 1, 1; -1, -1, 1; 0, 0, 1, 1; 0, 0, -1, -1, 1; 0, 0, 0, 0, 1, 1; 0, 0, 0, 0, -1, -1, 1; 0, 0, 0, 0, 0, 0, 1 ,1; 0, 0, 0, 0, 0, 0, -1, -1, 1; ... where the matrix inverse shifts columns left and up one place. ... The matrix log of A111940, with factorial denominators, begins: 0; 1/1!, 0; -1/2!, -1/1!, 0; 1/3!, 1/2!, 1/1!, 0; -2/4!, -1/3!, -1/2!, -1/1!, 0; 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0; 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0; 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; -2880/10!, -576/9!, -144/8!, -36/7!, -12/6!, -4/5!, -2/4!, -1/3!, -1/2!, -1/1!, 0; 14400/11!, 2880/10!, 576/9!, 144/8!, 36/7!, 12/6!, 4/5!, 2/4!, 1/3!, 1/2!, 1/1!, 0; ... Note that the square of the matrix log of A111940 begins: 0; 0, 0; -1, 0, 0; 0, -1, 0, 0; -1/12, 0, -1, 0, 0; 0, -1/12, 0, -1, 0, 0; -1/90, 0, -1/12, 0, -1, 0, 0; 0, -1/90, 0, -1/12, 0, -1, 0, 0; -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; -1/16632, 0, -1/3150, 0, -1/560, 0, -1/90, 0, -1/12, 0, -1, 0, 0; ... where nonzero terms are negative unit fractions with denominators given by A002544: [1, 12, 90, 560, 3150, 16632, 84084, 411840, ...,  C(2*n+1,n)*(n+1)^2, ...]. PROG (PARI) {T(n, k, q=-1) = local(A=Mat(1), B); if(n

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Last modified September 18 05:32 EDT 2019. Contains 327165 sequences. (Running on oeis4.)