reviewed

approved

proposed

reviewed

editing

proposed

G.f. for column 0: 1 = (1-3x) + 3*x/(1-2x)*(1-3x)(1-4x) + 15*x^2/(1-2x)^2*(1-3x)(1-4x)(1-5x) + 114*x^3/(1-2x)^3*(1-3x)(1-4x)(1-5x)(1-6x) + ... + T(n,0)*x^n/(1-2*x)^n*(1-3x)(1-4x)*..*(1-(n+3)x) + ...

15G.f. for column 1: 2 = 2*(1-4x) + 8*x/(1-2x)*(1-4x)(1-5x) + 56*x^2/(1-2x)^2*(1-3x4x)(1-5x)(1-6x) + 568*x^3/(1-2x)^3*(1-4x)(1-5x)(1-6x)(1-7x) + ... + T(n,1)*x^(n-1)/(1-2*x)^(n-1)*(1-4x)(1-5x) *..*(1-(n+3)x) + ...

114*x^3/(1-2x)^3*(1-3x)(1-4x)(1-5x)(1-6x) + ...

+ T(n,0)*x^n/(1-2*x)^n*(1-3x)(1-4x)*..*(1-(n+3)x) + ...

G.f. for column 1: 2 = 2*(1-4x) + 8*x/(1-2x)*(1-4x)(1-5x) +

56*x^2/(1-2x)^2*(1-4x)(1-5x)(1-6x) +

568*x^3/(1-2x)^3*(1-4x)(1-5x)(1-6x)(1-7x) + ...

+ T(n,1)*x^(n-1)/(1-2*x)^(n-1)*(1-4x)(1-5x)*..*(1-(n+3)x) + ...

proposed

editing

editing

proposed

Left-most Leftmost column is A082163 (enumerates acyclic automata with 2 inputs). The operation SHIFTUP(T) shifts each column of T up 1 row, dropping the elements that occupied the diagonal of T.

approved

editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 31 2005

G.f. for column k: T(k, k) = k+1 = Sum_{n>=k} T(n, k)*x^(n-k)/(1-2*x)^(n-k) * Product_{j=0..n-k} (1-(j+k+3)*x). Diagonalization: T = P*D*P^-1 where P(r, c) = A103247(r, c)/(r-c)! = (-1)^(r-c)*(c^2+2*c)^(r-c)/(r-c)! for r>=c>=1, and [P^-1](r, c) = A103242(r, c)/(r-c)!, and D is a diagonal matrix = {1, 2, 3, ...}.

nonn,tabl,new

nonn,tabl,new

Paul D . Hanna (pauldhanna(AT)juno.com), Jan 31 2005

G.f. for column k: T(k, k) = k+1 = Sum_{n>=k} T(n, k)*x^(n-k)/(1-2*x)^(n-k) * Product_{j=0..n-k} (1-(j+k+3)*x). Diagonalization: T = P*D*P^-1 where P(r, c) = A103247(r, c)/(r-c)! = (-1)^(r-c)*(c^2+2*c)^(r-c)/(r-c)! for r>=c>=1, and [P^-1](r, c) = A103242(r, c)/(r-c)!, and D is a diagonal matrix = {1, 2, 3, ...}.

nonn,tabl,new