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 A082158 Number of deterministic completely defined acyclic automata with 3 inputs and n transient labeled states (and a unique absorbing state). 3
 1, 1, 15, 1024, 198581, 85102056, 68999174203, 95264160938080, 207601975572545961, 674354204416939196800, 3122476748685067008205511, 19884561572783089348189507584, 169123749545536919971662851459485, 1874777145334671354828947023095675904, 26531967154935836079418311035871122812275 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This is the first column of the array A082170. LINKS V. A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003. V. A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551. FORMULA a(n)=a_3(n) where a_3(0) := 1, a_3(n) := sum(binomial(n, i)*(-1)^(n-i-1)*(i+1)^(3*n-3*i)*a_3(i), i=0..n-1), n>0. 1 = Sum_{n>=0} a(n)*exp(-(1+n)^3*x)*x^n/n!. - Vladeta Jovovic, Jul 18 2005 From Paul D. Hanna, May 03 2015: (Start) 1 = Sum_{n>=0} a(n) * x^n/(1 + (n+1)^3*x)^(n+1). 1 = Sum_{n>=0} a(n) * C(n+m-1,n) * x^n/(1 + (n+1)^3*x)^(n+m) for all m>=1. log(1+x) = Sum_{n>=1} a(n) * x^n/(1 + (n+1)^3*x)^n/n. (End) MATHEMATICA a[n_] := If[n == 0, 1, Sum[-(-1)^(n-k) Binomial[n, k] (k+1)^(3(n-k)) a[k], {k, 0, n-1}]]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Aug 29 2019 *) PROG (PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+(k+1)^3*x+x*O(x^n))^(k+1)), n)} for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, May 03 2015 (PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, -(-1)^(n-k)*binomial(n, k)*(k+1)^(3*(n-k))*a(k)))} for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, May 03 2015 CROSSREFS Cf. A082157, A082170. Sequence in context: A102102 A196569 A019282 * A064625 A241331 A205602 Adjacent sequences:  A082155 A082156 A082157 * A082159 A082160 A082161 KEYWORD easy,nonn AUTHOR Valery A. Liskovets, Apr 09 2003 EXTENSIONS More terms from Michel Marcus, Aug 29 2019 STATUS approved

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Last modified September 26 09:15 EDT 2021. Contains 347664 sequences. (Running on oeis4.)