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 A082164 Deterministic completely defined initially connected acyclic automata with 3 inputs and n+1 transient unlabeled states including a unique state having all transitions to the absorbing state. 2
 1, 7, 133, 5362, 380093, 42258384, 6830081860, 1520132414241, 447309239576913, 168599289097947589, 79364534944804317166, 45701029702436877135199, 31642128418550547009710906, 25960688434777959685891570936, 24926392120419324125117256758595, 27708074645788511889179577045508824 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Coefficients T_2(n,k) form the array A082172. These automata have no nontrivial automorphisms (by states). 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) := d_3(n)/(n-1)! where d_3(n) := b_3(n, 1)-sum(binomial(n-1, j-1)*T_3(n-j, j+1)*d_3(j), j=1..n-1); and T_3(0, k) := 1, T_3(n, k) := sum(binomial(n, i)*(-1)^(n-i-1)*((i+k+1)^3-1)^(n-i)*T_3(i, k), i=0..n-1), n>0. MATHEMATICA b[_, 0, _] = 1; b[k_, n_, r_] := b[k, n, r] = Sum[Binomial[n, t] (-1)^(n - t - 1) ((t + r + 1)^k - 1)^(n - t) b[k, t, r], {t, 0, n - 1}]; d3[n_] := d3[n] = b[3, n, 1] - Sum[Binomial[n - 1, j - 1] T3[n - j, j + 1] d3[j], {j, 1, n - 1}]; T3[0, _] = 1; T3[n_, k_] := T3[n, k] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + k + 1)^3 - 1)^(n - i) T3[i, k], {i, 0, n - 1}]; a[n_] := If[n == 1, 1, d3[n - 1]/(n - 2)!]; Array[a, 20] (* Jean-François Alcover, Aug 29 2019 *) CROSSREFS Cf. A082160, A082163, A082162. Sequence in context: A274788 A274258 A251577 * A229464 A317216 A119670 Adjacent sequences:  A082161 A082162 A082163 * A082165 A082166 A082167 KEYWORD easy,nonn AUTHOR Valery A. Liskovets, Apr 09 2003 EXTENSIONS More terms from Jean-François Alcover, Aug 29 2019 STATUS approved

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Last modified January 17 15:33 EST 2020. Contains 330958 sequences. (Running on oeis4.)