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A082158 Number of deterministic completely defined acyclic automata with 3 inputs and n transient labeled states (and a unique absorbing state). 3
1, 1, 15, 1024, 198581, 85102056, 68999174203, 95264160938080, 207601975572545961, 674354204416939196800, 3122476748685067008205511, 19884561572783089348189507584, 169123749545536919971662851459485, 1874777145334671354828947023095675904, 26531967154935836079418311035871122812275 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is the first column of the array A082170.

LINKS

Table of n, a(n) for n=0..14.

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)=a_3(n) where a_3(0) := 1, a_3(n) := sum(binomial(n, i)*(-1)^(n-i-1)*(i+1)^(3*n-3*i)*a_3(i), i=0..n-1), 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)

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-Fran├žois Alcover, Aug 29 2019 *)

PROG

(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

CROSSREFS

Cf. A082157, A082170.

Sequence in context: A102102 A196569 A019282 * A064625 A241331 A205602

Adjacent sequences:  A082155 A082156 A082157 * A082159 A082160 A082161

KEYWORD

easy,nonn

AUTHOR

Valery A. Liskovets, Apr 09 2003

EXTENSIONS

More terms from Michel Marcus, Aug 29 2019

STATUS

approved

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Last modified November 14 07:19 EST 2019. Contains 329111 sequences. (Running on oeis4.)