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,...}.

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

Offset: 0

Author

Paul D. Hanna, Jan 31 2005

Keywords

• nonn
• tabl

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.

Examples

			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],...
		

Crossrefs

Cf. A082164, A103248, A103243, A103236.

Programs

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
				

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, ...}.