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 A103237 Triangular matrix T, read by rows, that satisfies: T^3 + 3T^2 + 3T = SHIFTUP(T), also T^(n+2) + 3T^(n+1) + 3T^n = SHIFTUP(T^n - D*T^(n-1)) for all n, where D is a diagonal matrix with diagonal(D) = diagonal(T) = {1,2,3,...}. 2
 1, 7, 2, 133, 26, 3, 5362, 962, 63, 4, 380093, 66794, 3843, 124, 5, 42258384, 7380100, 409248, 11284, 215, 6, 6830081860, 1190206134, 65160081, 1709836, 27305, 342, 7, 1520132414241, 264665899160, 14416260516, 371199704, 5585270, 57798, 511, 8 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Leftmost column is A082164 (enumerates acyclic automata with 3 inputs). The operation SHIFTUP(T) shifts each column of T up 1 row, dropping the elements that occupied the diagonal of T. LINKS FORMULA T = P*D*P^-1 where P(r, c) = A103248(r, c)/(r-c)! = (-1)^(r-c)*(c^3+3*c^2+3*c)^(r-c)/(r-c)! for r>=c>=1 and [P^-1](r, c) = A103243(r, c)/(r-c)! and D is a diagonal matrix = {1, 2, 3, ...}. EXAMPLE Rows of T begin: [1], [7,2], [133,26,3], [5362,962,63,4], [380093,66794,3843,124,5], [42258384,7380100,409248,11284,215,6], [6830081860,1190206134,65160081,1709836,27305,342,7],... Rows of T^2 begin: [1], [21,4], [714,130,9], [41923,7410,441,16],... Rows of T^3 begin: [1], [49,8], [2821,494,27], [238238,41678,2331,64],... Rows of T^3 + 3*T^2 + 3*T equals SHIFTUP(T): [7], [133,26], [5362,962,63], [380093,66794,3843,124],... PROG (PARI) {T(n, k)=local(P, D); D=matrix(n+1, n+1, r, c, if(r==c, r)); P=matrix(n+1, n+1, r, c, if(r>=c, (-1)^(r-c)*(c^3+3*c^2+3*c)^(r-c)/(r-c)!)); return(if(n

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Last modified June 15 02:02 EDT 2021. Contains 345042 sequences. (Running on oeis4.)