A082159 Number of deterministic completely defined acyclic automata with 2 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.

1, 3, 39, 1206, 69189, 6416568, 881032059, 168514815360, 42934911510249, 14081311783382400, 5786296490491543599, 2914663547018935095552, 1767539279001227299807725, 1271059349855055258673975296, 1069996840045068513065229943875

Offset: 0

Author

Valery A. Liskovets, Apr 09 2003

Keywords

• easy
• nonn

Comments

This is the first column of the array A082171.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150
• 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.

Crossrefs

Cf. A082157, A082160, A082171.

Programs

Magma

function a(n) // a = A082159
  if n eq 0 then return 1;
  else return (&+[Binomial(n,j)*(-1)^(n-j-1)*((j+2)^2 - 1)^(n-j)*a(j): j in [0..n-1]]);
  end if;
end function;
[a(n): n in [0..20]]; // G. C. Greubel, Jan 17 2024

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + 2)^2 - 1)^(n - i) a[i], {i, 0, n - 1}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Aug 29 2019 *)

PARI

lista(nn)={my(a=vector(nn+1)); for(n=1, nn+1, a[n] = if(n==1, 1, sum(i=0, n-2, binomial(n-1, i)*(-1)^(n-i-2)*((i + 2)^2 - 1)^(n-i-1)*a[i+1]))); a;} \\ Petros Hadjicostas, Mar 07 2021

SageMath

@CachedFunction
def a(n): # A082159
    if n==0: return 1
    else: return sum(binomial(n,j)*(-1)^(n-j-1)*((j+2)^2 -1)^(n-j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024

Formula

a(n) = b_2(n), where b_2(0) = 1 and b_2(n) = Sum_{0..n-1} binomial(n, i) * (-1)^(n-i-1) * ((i + 2)^2 - 1)^(n-i) * b_2(i) for n > 0.