A103237 - OEIS

%I #11 Jul 20 2024 18:42:51

%S 1,7,2,133,26,3,5362,962,63,4,380093,66794,3843,124,5,42258384,

%T 7380100,409248,11284,215,6,6830081860,1190206134,65160081,1709836,

%U 27305,342,7,1520132414241,264665899160,14416260516,371199704,5585270,57798,511,8

%N 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,...}.

%C 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.

%F 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, ...}.

%e Rows of T begin:

%e [1],

%e [7,2],

%e [133,26,3],

%e [5362,962,63,4],

%e [380093,66794,3843,124,5],

%e [42258384,7380100,409248,11284,215,6],

%e [6830081860,1190206134,65160081,1709836,27305,342,7],...

%e Rows of T^2 begin:

%e [1],

%e [21,4],

%e [714,130,9],

%e [41923,7410,441,16],...

%e Rows of T^3 begin:

%e [1],

%e [49,8],

%e [2821,494,27],

%e [238238,41678,2331,64],...

%e Rows of T^3 + 3*T^2 + 3*T equals SHIFTUP(T):

%e [7],

%e [133,26],

%e [5362,962,63],

%e [380093,66794,3843,124],...

%o (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<k||k<0,0,(P*D*P^-1)[n+1,k+1]))}

%Y Cf. A082164, A103248, A103243, A103236.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Jan 31 2005