A103239 - OEIS

%I #10 Aug 30 2023 03:20:56

%S 1,2,8,52,480,5816,87936,1601728,34251520,843099616,23520367488,

%T 734404134336,25402332040704,964965390917120,39964015456707584,

%U 1793140743838290432,86691698782589288448,4494521175128812273152

%N Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.

%C a(n-1) = number of initially connected acyclic unlabeled n-state automata on a 2-letter input alphabet for which only one state is affected identically by both input letters. This state is necessarily one that is carried to the sink (absorbing state). For example, with n=2, a(1)=2 counts 2333, 3233, but not 2233. Here 1 is the source and 3 is the sink and 2333 is short for {{1, 2}, {1, 3}, {2, 3}, {2, 3}} giving the action of the input letters. The unlabeled condition is captured by requiring that the first appearances of 2,3,...,n occur in that order. A082161 counts these automata without the affected-identically restriction. - _David Callan_, Jun 07 2006

%H Vaclav Kotesovec, <a href="/A103239/b103239.txt">Table of n, a(n) for n = 0..32</a>

%F G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-x)^n*Product_{j=0..n} (1-(j+2)*x).

%e 1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) +

%e 52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...

%e + a(n)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...

%o (PARI) {a(n)=if(n<0,0,if(n==0,1,polcoeff( 1-sum(k=0,n-1,a(k)*x^k/(1-x)^k*prod(j=0,k,1-(j+2)*x+x*O(x^n))),n)))}

%Y Cf. A103238.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 31 2005