A082158 Number of deterministic completely defined acyclic automata with 3 inputs and n transient labeled states (and a unique absorbing state).

1, 1, 15, 1024, 198581, 85102056, 68999174203, 95264160938080, 207601975572545961, 674354204416939196800, 3122476748685067008205511, 19884561572783089348189507584, 169123749545536919971662851459485, 1874777145334671354828947023095675904, 26531967154935836079418311035871122812275

Offset: 0

Author

Valery A. Liskovets, Apr 09 2003

Keywords

• easy
• nonn

Comments

This is the first column of the array A082170.

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, A082159, A082160, A082170.

Programs

Magma

function a(n) // a = A082158
  if n eq 0 then return 1;
  else return (&+[Binomial(n,j)*(-1)^(n-j-1)*(j+1)^(3*n-3*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[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 *)

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

SageMath

@CachedFunction
def a(n): # A082158
    if n==0: return 1
    else: return sum(binomial(n,j)*(-1)^(n-j-1)*(j+1)^(3*n-3*j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024

Formula

a(n) = a_3(n) where a_3(0) = 1, a_3(n) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*(i+1)^(3*n-3*i)*a_3(i), 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)

Extensions

More terms from Michel Marcus, Aug 29 2019