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A103239 Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.

%I A103239 #10 Aug 30 2023 03:20:56
%S A103239 1,2,8,52,480,5816,87936,1601728,34251520,843099616,23520367488,
%T A103239 734404134336,25402332040704,964965390917120,39964015456707584,
%U A103239 1793140743838290432,86691698782589288448,4494521175128812273152
%N A103239 Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.
%C A103239 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 A103239 Vaclav Kotesovec, <a href="/A103239/b103239.txt">Table of n, a(n) for n = 0..32</a>
%F A103239 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-x)^n*Product_{j=0..n} (1-(j+2)*x).
%e A103239 1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) +
%e A103239 52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
%e A103239 + a(n)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
%o A103239 (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 A103239 Cf. A103238.
%K A103239 nonn
%O A103239 0,2
%A A103239 _Paul D. Hanna_, Jan 31 2005