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 A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1. 15
 1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 21, 5, 1, 3447, 1297, 461, 133, 31, 6, 1, 29093, 10063, 3447, 977, 225, 43, 7, 1, 273343, 87669, 29093, 8135, 1859, 353, 57, 8, 1, 2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1, 31998903 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column 0 equals A003319 (indecomposable permutations). Amazingly, column 1 (A104981) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A104986. From Paul D. Hanna, Feb 17 2009: (Start) Square array A156628 has columns found in this triangle T: Column 0 of A156628 = column 0 of T = A003319; Column 1 of A156628 = column 1 of T = A104981; Column 2 of A156628 = column 2 of T = A003319 shifted; Column 3 of A156628 = column 1 of T^2 (A104988); Column 5 of A156628 = column 2 of T^2 (A104988). (End) LINKS Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018. FORMULA T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+1, 2) = T(n, 0) for n>=0. EXAMPLE SHIFT_LEFT(column 0 of T^-1) = -1*(column 0 of T); SHIFT_LEFT(column 0 of T^1) = 1*(column 2 of T); SHIFT_LEFT(column 0 of T^2) = 2*(column 3 of T); where SHIFT_LEFT of column sequence shifts 1 place left. Triangle T begins: 1; 1,1; 3,2,1; 13,7,3,1; 71,33,13,4,1; 461,191,71,21,5,1; 3447,1297,461,133,31,6,1; 29093,10063,3447,977,225,43,7,1; 273343,87669,29093,8135,1859,353,57,8,1; 2829325,847015,273343,75609,17185,3251,523,73,9,1; ... Matrix inverse T^-1 is A104984 which begins: 1; -1,1; -1,-2,1; -3,-1,-3,1; -13,-3,-1,-4,1; -71,-13,-3,-1,-5,1; -461,-71,-13,-3,-1,-6,1; ... Matrix T also satisfies: [I + SHIFT_LEFT(T)] = [I - SHIFT_DOWN(T)]^-1, which starts: 1; 1,1; 2,1,1; 7,3,1,1; 33,13,4,1,1; 191,71,21,5,1,1; ... where SHIFT_DOWN(T) shifts columns of T down 1 row, and SHIFT_LEFT(T) shifts rows of T left 1 column, with both operations leaving zeros in the diagonal. MATHEMATICA T[n_, k_] := T[n, k] = If[n

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Last modified October 17 08:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)