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 A111544 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+3 of T), or [T^p](m,0) = p*T(p+m,p+3) for all m>=1 and p>=-3. 10
 1, 1, 1, 5, 2, 1, 33, 9, 3, 1, 261, 57, 15, 4, 1, 2361, 441, 99, 23, 5, 1, 23805, 3933, 783, 165, 33, 6, 1, 263313, 39249, 7083, 1383, 261, 45, 7, 1, 3161781, 430677, 71415, 13083, 2361, 393, 59, 8, 1, 40907241, 5137641, 789939, 136863, 23805, 3861, 567, 75, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column 0 equals A111530 (related to log of factorial series). Column 3 (A111547) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111549. 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+2, 2)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+3, 2) = 3*T(n+1, 0), T(n+4, 4) = T(n+1, 0), for n>=0. EXAMPLE SHIFT_LEFT(column 0 of T^-3) = -3*(column 0 of T); SHIFT_LEFT(column 0 of T^-2) = -2*(column 1 of T); SHIFT_LEFT(column 0 of T^-1) = -1*(column 2 of T); SHIFT_LEFT(column 0 of log(T)) = column 3 of T; SHIFT_LEFT(column 0 of T^1) = 1*(column 4 of T); where SHIFT_LEFT of column sequence shifts 1 place left. Triangle T begins: 1; 1,1; 5,2,1; 33,9,3,1; 261,57,15,4,1; 2361,441,99,23,5,1; 23805,3933,783,165,33,6,1; 263313,39249,7083,1383,261,45,7,1; 3161781,430677,71415,13083,2361,393,59,8,1; ... After initial term, column 2 is 3 times column 0. Matrix inverse T^-1 = A111548 starts: 1; -1,1; -3,-2,1; -15,-3,-3,1; -99,-15,-3,-4,1; -783,-99,-15,-3,-5,1; -7083,-783,-99,-15,-3,-6,1; ... where columns are all equal after initial terms; compare columns of T^-1 to column 2 of T. Matrix logarithm log(T) = A111549 is: 0; 1,0; 4,2,0; 23,6,3,0; 165,32,9,4,0; 1383,222,47,13,5,0; 13083,1824,321,70,18,6,0; ... compare column 0 of log(T) to column 3 of T. MATHEMATICA T[n_, k_] := T[n, k] = If[n

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Last modified January 28 15:20 EST 2022. Contains 350657 sequences. (Running on oeis4.)