A082162 Number of deterministic completely defined initially connected acyclic automata with 3 inputs and n transient unlabeled states (and a unique absorbing state).

1, 7, 139, 5711, 408354, 45605881, 7390305396, 1647470410551, 485292763088275, 183049273155939442, 86211400693272461866

Offset

1,2

Comments

Coefficients T_3(n,k) form the array A082170. These automata have no nontrivial automorphisms (by states).

References

R. Bacher, C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

Links

Vaclav Kotesovec (after Jean-François Alcover), Table of n, a(n) for n = 1..210

Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.

Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.

Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.

Formula

a(n) = c_3(n)/(n-1)! where c_3(n) = T_3(n, 1) - sum(binomial(n-1, j-1)*T_3(n-j, j+1)*c_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)^(3*n-3*i)*T_3(i, k), i=0..n-1), n>0.

Equals column 0 of triangle A102098. Also equals main diagonal of A102400: a(n) = A102098(n, 0) = A102400(n, n). - Paul D. Hanna, Jan 07 2005

Mathematica

T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[k == 0, 1, If[n == k, T[n, n-1], Sum[T[n-1, j]*(j+1)*((k+1)*(k+2)/2-j*(j+1)/2), {j, 0, k}]]]]; a[n_] := T[n, n]; Table[a[n], {n, 1, 11} ] (* Jean-François Alcover, Dec 15 2014 *)

Crossrefs

Cf. A082158, A082161.

Cf. A102098, A102400.

Sequence in context: A351334 A190195 A126156 * A280629 A348188 A238692

Adjacent sequences: A082159 A082160 A082161 * A082163 A082164 A082165

Keyword

easy,nonn,changed

Author

Valery A. Liskovets, Apr 09 2003

Extensions

More terms from Paul D. Hanna, Jan 07 2005

Status

approved