A082169 -id:A082169 - cp's OEIS frontend

A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

1, 3, 16, 127, 1363, 18628, 311250, 6173791, 142190703, 3737431895, 110577492346, 3641313700916, 132214630355700, 5251687490704524, 226664506308709858, 10568175957745041423, 529589006347242691143, 28395998790096299447521

Offset: 1

Valery A. Liskovets, Apr 09 2003

Coefficients T_2(n,k) form the array A082169. These automata have no nontrivial automorphisms (by states).

Also counts the relaxed compacted binary trees of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

			a(2)=3 since the following transition diagrams represent all three initially connected acyclic automata with two input letters x and y, two transient states 1 (initial) and 2 and the absorbing state 0:
  1 == x, y==> 2 == x, y ==> 0 == x, y ==> 0, 1 -- x --> 2 == x, y ==> 0 == x, y ==> 0
  1 -- y --> 0
and the last one with x and y interchanged.

Roland Bacher and Christophe Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
David Callan, A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata, arXiv:0704.0004 [math.CO], 2007.
Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
Andrew Elvey Price, Wenjie Fang, and Michael Wallner, Compacted binary trees admit a stretched exponential, arXiv:1908.11181 [math.CO], 2019-2020; J. Combin. Theory Ser. A 177 (2021), Paper No. 105306, 40 pp.
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic enumeration of compacted binary trees of bounded right height, arXiv:1703.10031 [math.CO], 2017; J. Combin. Theory Ser. A 172 (2020), 105177, 49 pp.
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.
Fedor Petrov, On a generating function and vector ν of length n, answer to question on MathOverflow (2024).
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018; Theoret. Comput. Sci. 755 (2019), 1-12.

Cf. A008276, A082157, A082169, A102086, A102316, A254789.

a[n_]:= a[n]= If[n==0, 1, Coefficient[1-Sum[a[k]*x^k*Product[1-j*x, {j, 1, k+1}], {k, 0, n-1}], x, n]];
Table[a[n], {n, 18}] (* Jean-François Alcover, Dec 15 2014, after Paul D. Hanna *)

{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k+1,1-j*x+x*O(x^n))),n))} \\ Paul D. Hanna, Jan 07 2005

{a(n)=local(A);if(n<1,0,A=x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(1-prod(i=1,k+1,1-i*x))); polcoeff(A,n))} /* Michael Somos, Jan 16 2005 */

upto(n) = my(v=vector(n+1, i, i==1)); for(i=1, n, for(j=i+1, n+1, v[j] += i*v[j-1])); v[2..#v] \\ Mikhail Kurkov, Oct 25 2024

from functools import cache
@cache
def b(n, k):
    if n == 0: return k + 1
    return sum(b(j, k)*b(n-j-1, k+j) for j in range(n))
def A082161(n): return b(n, 0)
print([A082161(n) for n in range(1, 19)]) # G. C. Greubel, Jan 18 2024

a(n) = c_2(n)/(n-1)! where c_2(n) = T_2(n, 1) - Sum_{j=1..n-1} binomial(n-1, j-1)*T_2(n-j, j+1)*c_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)^(2*n-2*i)*T_2(i, k), n > 0.
Equals column 0 of triangle A102086. Also equals main diagonal of A102316: a(n) = A102086(n, 0) = A102316(n, n). - Paul D. Hanna, Jan 07 2005
G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=1, n+1} (1-k*x) for n>0 with a(0)=1. a(n) = -Sum_{k=1, [(n+1)/2]} A008276(n-k+1, k)*a(n-k) where A008276 is Stirling numbers of the first kind. Thus G.f.: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + a(n)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ... with a(0)=1. - Paul D. Hanna, Jan 14 2005
a(n) is the determinant of the n X n matrix with (i, j) entry = StirlingCycle[i+1, 2i-j]. - David Callan, Jul 20 2005
a(n) = b(n,0) where b(0,p) = p+1 and b(n+1,p) = Sum_{i=0..n} b(i,p)*b(n-i,p+i) for n>=1. - Michael Wallner, Apr 20 2017
From Michael Wallner, Jan 31 2022: (Start)
a(n) = r(n,n) where r(n,m)=(m+1)*r(n-1,m)+r(n,m-1) for n>=m>=1, r(n,m)=0 for n=0.
a(n) = Theta(n!*4^n*exp(3*a1*n^(1/3))*n) for large n, where a1=-2.338... is the largest root of the Airy function Ai(x) of the first kind; see [Elvey Price, Fang, Wallner 2021]. (End)

A082157 Number of deterministic completely defined acyclic automata with 2 inputs and n transient labeled states (and a unique absorbing state).

Original entry on oeis.org

1, 1, 7, 142, 5941, 428856, 47885899, 7685040448, 1681740027657, 482368131521920, 175856855224091311, 79512800815739448576, 43701970591391787395197, 28714779850695689959247872

Offset: 0

Valery A. Liskovets, Apr 09 2003

This is the first column of the array A082169.

			a(2)=7 since the following transition diagrams represent all seven acyclic automata with two input letters x and y, two transient states 1 and 2, and the absorbing state 0:
1==x,y==>0==x,y==>0<==x,y==2, 1==x,y==>2==x,y==>0==x,y==>0,
the same with 1 and 2 interchanged,
1--x-->2==x,y==>0==x,y==>0
1--y-->0
and the last one with x and y and/or 1 and 2 interchanged.

Vaclav Kotesovec, Table of n, a(n) for n = 0..220
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.

a[0] = 1; a[n_] := a[n] = Sum[-(-1)^(n-k)*Binomial[n, k]*(k+1)^(2*(n-k))*a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 15 2014, translated from PARI *)

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+(k+1)^2*x+x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna

{a(n)=if(n==0,1,sum(k=0,n-1,-(-1)^(n-k)*binomial(n,k)*(k+1)^(2*(n-k))*a(k)))}

a(n) = a_2(n) where a_2(0) := 1, a_2(n) := sum(binomial(n, i)*(-1)^(n-i-1)*(i+1)^(2*n-2*i)*a_2(i), i=0..n-1), n>0.
1 = Sum_{n>=0} a(n)*exp(-(1+n)^2*x)*x^n/n!. - Vladeta Jovovic, Jul 18 2005
From Paul D. Hanna, Jun 04 2011: (Start)
1 = Sum_{n>=0} a(n)*x^n/(1 + (n+1)^2*x)^(n+1).
1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + (n+1)^2*x)^(n+m) for m>=1.
log(1+x) = Sum_{n>=1} a(n)*x^n/(1 + (n+1)^2*x)^n/n. (End)

A082170 Deterministic completely defined quasi-acyclic automata with 3 inputs, n transient and k absorbing labeled states.

Original entry on oeis.org

1, 1, 1, 1, 8, 15, 1, 27, 368, 1024, 1, 64, 2727, 53672, 198581, 1, 125, 11904, 710532, 18417792, 85102056, 1, 216, 38375, 4975936, 386023509, 12448430408, 68999174203, 1, 343, 101520, 23945000, 3977848832, 381535651512, 14734002979456, 95264160938080

Offset: 0

Valery A. Liskovets, Apr 09 2003

Array read by antidiagonals: (0,1),(0,2),(1,1),(0,3),...

The first column is A082158.

			The array begins:
            1,              1,               1,             1, ...;
            1,              8,              27,            64, ...;
           15,            368,            2727,         11904, ...;
         1024,          53672,          710532,       4975936, ...;
       198581,       18417792,       386023509,    3977848832, ...;
     85102056,    12448430408,    381535651512, 5451751738944, ...;
  68999174203, 14734002979456, 624245820664563, ...;
Antidiagonals begin as:
  1;
  1,   1;
  1,   8,    15;
  1,  27,   368,    1024;
  1,  64,  2727,   53672,    198581;
  1, 125, 11904,  710532,  18417792,    85102056;
  1, 216, 38375, 4975936, 386023509, 12448430408, 68999174203;

G. C. Greubel, Antidiagonals n = 0..50, flattened
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.

Cf. A082158, A082162, A082169.

function A(n,k)
  if n eq 0 then return 1;
  else return (&+[(-1)^(n-j+1)*Binomial(n,j)*(k+j)^(3*n-3*j)*A(j,k): j in [0..n-1]]);
  end if;
end function;
A082170:= func< n,k | A(k,n-k+1) >;
[A082170(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 19 2024

T[0, ] = 1; T[n, k_]:= T[n, k] = Sum[Binomial[n, i] (-1)^(n-i-1)*(i + k)^(3n-3i) T[i, k], {i,0,n-1}];
Table[T[n-k-1, k], {n, 1, 9}, {k, n-1, 1, -1}]//Flatten (* Jean-François Alcover, Aug 29 2019 *)

@CachedFunction
def A(n,k):
    if n==0: return 1
    else: return sum((-1)^(n-j+1)*binomial(n,j)*(k+j)^(3*n-3*j)*A(j,k) for j in range(n))
def A082170(n,k): return A(k,n-k+1)
flatten([[A082170(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jan 19 2024

T(n, k) = T_3(n, k) where T_3(0, k) = 1, T_3(n, k) = Sum_{i=0..n-1} (-1)^(n-i-1)*binomial(n, i)*(i+k)^(3*n-3*i)*T_3(i, k), n > 0.

A082171 A subclass of quasi-acyclic automata with 2 inputs, n transient and k absorbing labeled states; square array T(n,k) read by descending antidiagonals (n >= 0 and k >= 1).

Original entry on oeis.org

1, 1, 3, 1, 8, 39, 1, 15, 176, 1206, 1, 24, 495, 7784, 69189, 1, 35, 1104, 29430, 585408, 6416568, 1, 48, 2135, 84600, 2791125, 67481928, 881032059, 1, 63, 3744, 204470, 9841728, 389244600, 11111547520, 168514815360, 1, 80, 6111, 437616, 28569765, 1627740504, 75325337235, 2483829653544, 42934911510249

Offset: 0

Valery A. Liskovets, Apr 09 2003

Array read by descending antidiagonals: (0,1), (0,2), (1,1), (0,3), ...

The first column is A082159; i.e., T(n,k=1) = A082159(n). [The number n of transient states in the name of square array T(n,k) does not include the pre-dead transient state, which is, however, included in the name of A082159. See Section 3.1 in Liskovets (2006). - Petros Hadjicostas, Mar 07 2021]

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins:
          1,           1,           1,          1,        1, ...;
          3,           8,          15,         24,       35, ...;
         39,         176,         495,       1104,     2135, ...;
       1206,        7784,       29430,      84600,   204470, ...;
      69189,      585408,     2791125,    9841728, 28569765, ...;
    6416568,    67481928,   389244600, 1627740504, ...;
  881032059, 11111547520, 75325337235, ...;
  ...
Triangular array A(n,k) = T(k-1, n-k+1) (with rows n >= 1 and columns k = 1..n), read from the antidiagonals downwards of square array T:
  1;
  1,  3,
  1,  8,   39;
  1, 15,  176,  1206;
  1, 24,  495,  7784,   69189;
  1, 35, 1104, 29430,  585408,  6416568;
  1, 48, 2135, 84600, 2791125, 67481928, 881032059;
  ...

G. C. Greubel, Antidiagonals n = 0..50, flattened
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.

Cf. A082159, A082163, A082169.

function A(n,k)
  if n eq 0 then return 1;
  else return (&+[(-1)^(n-j+1)*Binomial(n,j)*((k+j+1)^2-1)^(n-j)*A(j,k): j in [0..n-1]]);
  end if;
end function;
A082171:= func< n,k | A(k,n-k+1) >;
[A082171(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 19 2024

T[0, ] = 1; T[n, k_] := T[n, k] = Sum[Binomial[n, i] (-1)^(n - i - 1)*((i + k + 1)^2 - 1)^(n - i)*T[i, k], {i, 0, n - 1}];
Table[T[n - k - 1, k], {n, 1, 10}, {k, n - 1, 1, -1}] // Flatten (* Jean-François Alcover, Aug 29 2019 *)

lista(nn,kk)={my(T=matrix(nn+1,kk)); for(n=1, nn+1, for(k=1, kk, 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])))); T;} \\ Petros Hadjicostas, Mar 07 2021

@CachedFunction
def A(n,k):
    if n==0: return 1
    else: return sum((-1)^(n-j+1)*binomial(n,j)*((k+j+1)^2-1)^(n-j)*A(j,k) for j in range(n))
def A082171(n,k): return A(k,n-k+1)
flatten([[A082171(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jan 19 2024

T(n, k) = S_2(n, k) where S_2(0, k) := 1 and S_2(n, k) := Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*((i + k + 1)^2 - 1)^(n-i)*S_2(i, k) for n > 0.

Name clarified by Petros Hadjicostas, Mar 07 2021

A103240 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2)^(n-k)/(n-k)! for n >= k >= 1.

Original entry on oeis.org

1, 1, 1, 7, 4, 1, 142, 56, 9, 1, 5941, 1780, 207, 16, 1, 428856, 103392, 9342, 544, 25, 1, 47885899, 9649124, 709893, 32848, 1175, 36, 1, 7685040448, 1329514816, 82305144, 3142528, 91150, 2232, 49, 1, 1681740027657, 254821480596, 13598786979

Offset: 1

Paul D. Hanna, Feb 02 2005

Define a triangular matrix P where P(n,k) = (-k^2)^(n-k)/(n-k)!; then M = P*D*P^-1 = A102086 satisfies M^2 = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row. Essentially equal to square array A082169 as a triangular matrix. The first column is A082157 (enumerates acyclic automata with 2 inputs).

			Rows of unreduced fractions T(n,k)/(n-k)! begin:
  [1/0!],
  [1/1!, 1/0!],
  [7/2!, 4/1!, 1/0!],
  [142/3!, 56/2!, 9/1!, 1/0!],
  [5941/4!, 1780/3!, 207/2!, 16/1!, 1/0!],
  [428856/5!, 103392/4!, 9342/3!, 544/2!, 25/1!, 1/0!],
  [47885899/6!, 9649124/5!, 709893/4!, 32848/3!, 1175/2!, 36/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103245(n,k)/(n-k)!:
  [1/0!],
  [-1/1!, 1/0!],
  [1/2!, -4/1!, 1/0!],
  [-1/3!, 16/2!, -9/1!, 1/0!],
  [1/4!, -64/3!, 81/2!, -16/1!, 1/0!], ...

Cf. A103245, A102086, A082169, A082157.

{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^2)^(r-c)/(r-c)!))); return(if(n

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2)^(n-m)*T(m, k).

For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2)^(j-k)*T(n, j).

cp's OEIS Frontend

A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

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A082157 Number of deterministic completely defined acyclic automata with 2 inputs and n transient labeled states (and a unique absorbing state).

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A082170 Deterministic completely defined quasi-acyclic automata with 3 inputs, n transient and k absorbing labeled states.

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A082171 A subclass of quasi-acyclic automata with 2 inputs, n transient and k absorbing labeled states; square array T(n,k) read by descending antidiagonals (n >= 0 and k >= 1).

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A103240 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2)^(n-k)/(n-k)! for n >= k >= 1.

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