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A103239
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Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.
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1
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1, 2, 8, 52, 480, 5816, 87936, 1601728, 34251520, 843099616, 23520367488, 734404134336, 25402332040704, 964965390917120, 39964015456707584, 1793140743838290432, 86691698782589288448, 4494521175128812273152
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (callan(AT)stat.wisc.edu), Jun 07 2006
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FORMULA
| G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-x)^n*Product_{j=0..n} (1-(j+2)*x).
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EXAMPLE
| 1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) +
52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
+ a(n)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
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PROG
| (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)))}
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CROSSREFS
| Cf. A103238.
Sequence in context: A006351 A089467 A195192 * A132228 A151879 A191552
Adjacent sequences: A103236 A103237 A103238 * A103240 A103241 A103242
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2005
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