A082163 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n+1 transient unlabeled states including a unique state having all transitions to the absorbing state.

1, 3, 15, 114, 1191, 15993, 263976, 5189778, 118729335, 3104549229, 91472523339, 3002047651764, 108699541743348, 4307549574285900, 185545521930558012, 8636223446937857130, 432133295481763698951, 23140414627731672497973, 1320835234697505382760757, 80076275471464881277266666, 5139849930933791535446756127

Offset: 1

Author

Valery A. Liskovets, Apr 09 2003

Keywords

• easy
• nonn

Comments

Coefficients T_2(n,k) form the array A082171. These automata have no nontrivial automorphisms (by states).

Also equals the leftmost column of triangular matrix M=A103236, which satisfies: M^2 + 2*M = SHIFTUP(M) (i.e. each column of M shifts up 1 row). - Paul D. Hanna, Jan 29 2005

Links

• Valery A. Liskovets, The number of connected initial automata, Kibernetika (Kiev), 3 (1969), 16-19 (in Russian; English translation: Cybernetics, 4 (1969), 259-262). [It includes the original methodology that he used in his 2003 and 2006 papers.]
• 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. A082159, A082161, A082171, A103236.

Programs

Mathematica

a[n_] := a[n] = If[n<1, 0, If[n == 1, 1, SeriesCoefficient[1-Sum[a[k+1]*x^k/(1-2*x)^k*Product[1-(j+3)*x, {j, 0, k}], {k, 0, n-2}], {x, 0,
n-1}]]]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 15 2014, after PARI *)

PARI

{a(n)=if(n<1,0,if(n==1,1,polcoeff( 1-sum(k=0,n-2,a(k+1)*x^k/(1-2*x)^k*prod(j=0,k,1-(j+3)*x+x*O(x^n))),n-1)))} \\ Paul D. Hanna, Jan 29 2005
/* Second PARI program using Valery A. Liskovets's recurrence: */
lista(nn)={my(T=matrix(nn+1, nn+1)); my(d=vector(nn)); my(a=vector(nn)); for(n=1, nn+1, for(k=1, nn, T[n, k] = if(n==1, 1, sum(i=0, n-2, binomial(n-1, i)*(-1)^(n-i-2)*((i + k + 1)^2 - 1)^(n-i-1)*T[i+1, k])))); for(n=1, nn, d[n] = T[n+1,1] - sum(j=1, n-1, binomial(n-1, j-1)*T[n-j+1, j+1]*d[j])); for(n=1, nn, a[n] = if(n==1, 1, d[n-1]/(n-2)!)); a;} \\ Petros Hadjicostas, Mar 07 2021

Formula

a(1) = 1 and a(n) := d_2(n-1)/(n-2)! for n >= 2, where d_2(n) := T_2(n, 1) - Sum_{j=1..n-1} binomial(n-1, j-1) * T_2(n-j, j+1) * d_2(j); and T_2(0, k) := 1, T_2(n, k) := Sum_{i=0..n-1} binomial(n, i) * (-1)^(n-i-1) *((i+k+1)^2 - 1)^(n-i) * T_2(i, k) for n > 0. [Edited by Petros Hadjicostas, Mar 06 2021 to agree with Theorem 3.3 (p. 543) in Liskovets (2006). Here, n + 1 is "the number of transient states including the pre-dead state".]

G.f.: 1 = Sum_{n>=0} a(n+1) * (x^n/(1-2*x)^n) * Product_{k=0..n} (1 - (3 + k)*x). Thus: 1 = 1*(1-3x) + 3*(x/(1-2x))*(1-3x)*(1-4x) + 15*(x^2/(1-2x)^2)*(1-3x)*(1-4x)*(1-5x) + 114*(x^3/(1-2x)^3)*(1-3x)*(1-4x)*(1-5x)*(1-6x) + ... - Paul D. Hanna, Jan 29 2005

Extensions

More terms from Petros Hadjicostas, Mar 06 2021 using the above programs