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A082160 Deterministic completely defined acyclic automata with 3 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.

%I A082160 #19 Jan 20 2024 09:26:50
%S A082160 1,7,315,45682,15646589,10567689552,12503979423607,23841011541867520,
%T A082160 68835375121428936153,286850872894190847235840,
%U A082160 1660638682341609286358474579,12947089879912710544534553836032
%N A082160 Deterministic completely defined acyclic automata with 3 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.
%C A082160 This is the first column of the array A082172.
%H A082160 G. C. Greubel, <a href="/A082160/b082160.txt">Table of n, a(n) for n = 0..150</a>
%H A082160 Valery A. Liskovets, <a href="http://igm.univ-mlv.fr/~fpsac/FPSAC03/ARTICLES/5.pdf">Exact enumeration of acyclic automata</a>, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
%H A082160 Valery A. Liskovets, <a href="http://dx.doi.org/10.1016/j.dam.2005.06.009">Exact enumeration of acyclic deterministic automata</a>, Discrete Appl. Math., 154, No.3 (2006), 537-551.
%F A082160 a(n) = b_3(n) where b_3(0) = 1, b_3(n) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*((i+2)^3 - 1)^(n-i)*b_3(i), n > 0.
%t A082160 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + 2)^3 - 1)^(n - i) a[i], {i, 0, n - 1}];
%t A082160 Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Aug 29 2019 *)
%o A082160 (Magma)
%o A082160 function a(n) // a = A082160
%o A082160   if n eq 0 then return 1;
%o A082160   else return (&+[Binomial(n,j)*(-1)^(n-j-1)*((j+2)^3 - 1)^(n-j)*a(j): j in [0..n-1]]);
%o A082160   end if;
%o A082160 end function;
%o A082160 [a(n): n in [0..20]]; // _G. C. Greubel_, Jan 17 2024
%o A082160 (SageMath)
%o A082160 @CachedFunction
%o A082160 def a(n): # A082160
%o A082160     if n==0: return 1
%o A082160     else: return sum(binomial(n,j)*(-1)^(n-j-1)*((j+2)^3 -1)^(n-j)*a(j) for j in range(n))
%o A082160 [a(n) for n in range(21)] # _G. C. Greubel_, Jan 17 2024
%Y A082160 Cf. A082159, A082158, A082172.
%K A082160 easy,nonn
%O A082160 0,2
%A A082160 _Valery A. Liskovets_, Apr 09 2003