A082157 -id:A082157 - 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)

A082169 Deterministic completely defined quasi-acyclic automata with 2 inputs, n transient and k absorbing labeled states.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 1, 9, 56, 142, 1, 16, 207, 1780, 5941, 1, 25, 544, 9342, 103392, 428856, 1, 36, 1175, 32848, 709893, 9649124, 47885899, 1, 49, 2232, 91150, 3142528, 82305144, 1329514816, 7685040448, 1, 64, 3871, 215892, 10682325, 440535696, 13598786979, 254821480596, 1681740027657

Offset: 0

Valery A. Liskovets, Apr 09 2003

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

The first column is A082157.

			The array begins:
         1,          1,           1,           1,          1, ...;
         1,          4,           9,          16,         25, ...;
         7,         56,         207,         544,       1175, ...;
       142,       1780,        9342,       32848,      91150, ...;
      5941,     103392,      709893,     3142528,   10682325, ...;
    428856,    9649124,    82305144,   440535696, 1775027000, ...;
  47885899, 1329514816, 13598786979, 85529171200, ...;
Antidiagonal triangle begins as:
  1;
  1,  1;
  1,  4,    7;
  1,  9,   56,   142;
  1, 16,  207,  1780,    5941;
  1, 25,  544,  9342,  103392,   428856;
  1, 36, 1175, 32848,  709893,  9649124,   47885899;
  1, 49, 2232, 91150, 3142528, 82305144, 1329514816, 7685040448;

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. A082157, A082161.

function A(n,k)
  if n eq 0 then return 1;
  else return (&+[(-1)^(n-j+1)*Binomial(n,j)*(k+j)^(2*n-2*j)*A(j,k): j in [0..n-1]]);
  end if;
end function;
A082169:= func< n,k | A(k,n-k+1) >;
[A082169(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)^(2n-2i) 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 *)

@CachedFunction
def A(n,k):
    if n==0: return 1
    else: return sum((-1)^(n-j+1)*binomial(n,j)*(k+j)^(2*n-2*j)*A(j,k) for j in range(n))
def A082169(n,k): return A(k,n-k+1)
flatten([[A082169(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jan 19 2024

T(n, k) = T_2(n, k) where T_2(0, k) = 1, T_2(n, k) = Sum_{i=0..n-1} (-1)^(n-i-1)*binomial(n, i)*(i+k)^(2*(n-i))*T_2(i, k), n > 0.

A082159 Number of deterministic completely defined acyclic automata with 2 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.

Original entry on oeis.org

1, 3, 39, 1206, 69189, 6416568, 881032059, 168514815360, 42934911510249, 14081311783382400, 5786296490491543599, 2914663547018935095552, 1767539279001227299807725, 1271059349855055258673975296, 1069996840045068513065229943875

Offset: 0

Valery A. Liskovets, Apr 09 2003

This is the first column of the array A082171.

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.

Cf. A082157, A082160, A082171.

function a(n) // a = A082159
  if n eq 0 then return 1;
  else return (&+[Binomial(n,j)*(-1)^(n-j-1)*((j+2)^2 - 1)^(n-j)*a(j): j in [0..n-1]]);
  end if;
end function;
[a(n): n in [0..20]]; // G. C. Greubel, Jan 17 2024

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + 2)^2 - 1)^(n - i) a[i], {i, 0, n - 1}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Aug 29 2019 *)

lista(nn)={my(a=vector(nn+1)); for(n=1, nn+1, a[n] = if(n==1, 1, sum(i=0, n-2, binomial(n-1, i)*(-1)^(n-i-2)*((i + 2)^2 - 1)^(n-i-1)*a[i+1]))); a;} \\ Petros Hadjicostas, Mar 07 2021

@CachedFunction
def a(n): # A082159
    if n==0: return 1
    else: return sum(binomial(n,j)*(-1)^(n-j-1)*((j+2)^2 -1)^(n-j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024

a(n) = b_2(n), where b_2(0) = 1 and b_2(n) = Sum_{0..n-1} binomial(n, i) * (-1)^(n-i-1) * ((i + 2)^2 - 1)^(n-i) * b_2(i) for n > 0.

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

Original entry on oeis.org

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

Offset: 0

Valery A. Liskovets, Apr 09 2003

This is the first column of the array A082170.

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.

Cf. A082157, A082159, A082160, A082170.

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

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 *)

{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

{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

@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

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)

More terms from Michel Marcus, Aug 29 2019

A195736 E.g.f.: x = Sum_{n>=1} a(n)x^n/n! exp(-n^2*x).

Original entry on oeis.org

1, 2, 21, 568, 29705, 2573136, 335201293, 61480323584, 15135660248913, 4823681315219200, 1934425407465004421, 954153609788873382912, 568125617688093236137561, 402006917909739659429470208, 333597313002114320208678928125

Offset: 1

Paul D. Hanna, Sep 30 2011

nonn

Compare e.g.f. to: x = Sum_{n>=1} n^(n-1)*x^n/n! * exp(-n*x), which generates coefficients for the series reversion of x*exp(-x).

			x = x*exp(-x) + 2*x^2/2!*exp(-4*x) + 21*x^3/3!*exp(-9*x) + 568*x^4/4!*exp(-16*x) + 29705*x^5/5!*exp(-25*x) +...+ a(n)*x^n/n!*exp(-n^2*x) +...
The coefficients a(n) also satisfy:
x = x/(1+x) + 2*x^2/(2*(1+4*x)^2) + 21*x^3/(3*(1+9*x)^3) + 568*x^4/(4*(1+16*x)^4) + 29705*x^5/(5*(1+25*x)^5) +...+ a(n)*x^n/(n*(1+n^2*x)^n) +...

Cf. A195737, A082157.

{a(n)=if(n<1,0,n!*polcoeff(x-sum(m=1,n-1,a(m)*x^m/m!*exp(-m^2*x+x*O(x^n))),n))}

{a(n)=if(n<1,0,n*polcoeff(x-sum(m=1,n-1,a(m)*x^m/(m*(1+m^2*x+x*O(x^n))^m)),n))}

G.f.: x = Sum_{n>=1} a(n)*x^n/(n*(1 + n^2*x)^n).

a(n) = n*A082157(n+1).

A196304 G.f.: x = Sum_{n>=1} a(n)x^n/(1 + n(n+1)/2*x)^n.

Original entry on oeis.org

1, 1, 5, 64, 1587, 65421, 4071178, 357962760, 42379107165, 6512954469625, 1262574678261816, 301690485704179584, 87187147717429037215, 29994563760476311689525, 12119686846920536310216000, 5685713204308826743851247936, 3066004482905684870319668989977

Offset: 1

Paul D. Hanna, Sep 30 2011

nonn

Compare g.f. to: x = Sum_{n>=1} n^(n-2)*x^n/(1 + n*x)^n, which generates coefficients in the series reversion of x*exp(-x).

			x = x/(1+x) + 1*x^2/(1+3*x)^2 + 5*x^3/(1+6*x)^3 + 64*x^4/(1+10*x)^4 + 1587*x^5/(1+15*x)^5 +...+ a(n)*x^n/(1+n*(n+1)/2*x)^n +...
The coefficients a(n) also satisfy:
x = x*exp(-x) + 1*x^2/1!*exp(-3*x) + 5*x^3/2!*exp(-6*x) + 64*x^4/3!*exp(-10*x) + 1587*x^5/4!*exp(-15*x) +...+ a(n)*x^n/(n-1)!*exp(-n*(n+1)/2*x) +...

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A195737, A082157.

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
a:= n-> p([i*(i+1)/2$i=1..n-1]):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 03 2015

{a(n)=if(n<1, 0, polcoeff(x-sum(m=1, n-1, a(m)*x^m/(1+m*(m+1)/2*x+x*O(x^n))^m), n))}

{a(n)=if(n<1, 0, (n-1)!*polcoeff(x-sum(m=1, n-1, a(m)*x^m/(m-1)!*exp(-m*(m+1)/2*x+x*O(x^n))), n))}

E.g.f.: x = Sum_{n>=1} a(n)*x^n/(n-1)! * exp(-n*(n+1)/2*x).
a(n) = A195737(n)/n for n>=1.
a(n) = Sum_{k=1..n-1} (-1)^(k-1)*binomial(n-1,k)*binomial(n+1-k,2)^k*a(n-k) for n>=2. - Jonathan Noel, May 05 2017

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).

A229806 G.f.: Sum_{n>=1} a(n)x^n / (1 + nx)^(n^2) = x.

Original entry on oeis.org

1, 1, 7, 150, 6924, 569726, 74358042, 14229990742, 3774315375580, 1330122245198910, 602741550311798067, 342138788139339603446, 238146938124253555981224, 199695655908033678248780110, 198741234873020798204357773510, 231773141251670398730627959107510

Offset: 1

Paul D. Hanna, Oct 02 2013

nonn

Compare to identity: Sum_{n>=1} n^(n-2) * x^n / (1 + n*x)^n = x.

			G.f.: x = 1*x/(1+x) + 1*x^2/(1+2*x)^4 + 7*x^3/(1+3*x)^9 + 150*x^4/(1+4*x)^16 + 6924*x^5/(1+5*x)^25 + 569726*x^6/(1+6*x)^36 +...

Cf. A177447, A082157.

{a(n)=polcoeff(x-sum(k=1, n-1, a(k)*x^k/(1+k*x+x*O(x^n))^(k^2)), n)}
for(n=1,20,print1(a(n),", "))

A275763 G.f.: x = Sum_{n>=1} a(n-1) * x^n / Product_{k=1..n} (1 + nkx).

Original entry on oeis.org

1, 1, 5, 63, 1514, 59685, 3512620, 289374295, 31846112564, 4518890895645, 804124456255680, 175478742025495755, 46106223230016643056, 14363471037818609599893, 5236804141734580288633760, 2209636417728549950873988255, 1068573377399399933312154968064, 587247047578198565707709826415149, 364003505996839798561347571968317760, 252786592402514515785220127177096089395

Offset: 0

Paul D. Hanna, Aug 18 2016

nonn

Compare g.f. to:
(1) x = Sum_{n>=1} (n-1)! * x^n / Product_{k=1..n} (1 + k*x).
(2) x = Sum_{n>=1} n^(n-2) * x^n / (1 + n*x)^n.
(3) x = Sum_{n>=1} (n-1)!^2 * x^n / Product_{k=1..n} (1 + k^2*x).
(4) x = Sum_{n>=1} A082157(n-1) * x^n / (1 + n^2*x)^n.

			G.f.: x = 1*x/(1+x) + 1*x^2/((1+2*1*x)*(1+2*2*x)) + 5*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 63*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) + 1514*x^5/((1+5*1*x)*(1+5*2*x)*(1+5*3*x)*(1+5*4*x)*(1+5*5*x)) + 59685*x^6/((1+6*1*x)*(1+6*2*x)*(1+6*3*x)*(1+6*4*x)*(1+6*5*x)*(1+6*6*x)) +...

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = -Vec(sum(m=1,#A, A[m]*x^m/prod(k=1,m,(1 + m*k*x +x*O(x^#A) ) ) ) )[#A] );A[n+1]}
for(n=0,30,print1(a(n),", "))

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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A082169 Deterministic completely defined quasi-acyclic automata with 2 inputs, n transient and k absorbing labeled states.

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A082159 Number of deterministic completely defined acyclic automata with 2 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.

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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).

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A195736 E.g.f.: x = Sum_{n>=1} a(n)*x^n/n! * exp(-n^2*x).

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A196304 G.f.: x = Sum_{n>=1} a(n)*x^n/(1 + n*(n+1)/2*x)^n.

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A195736 E.g.f.: x = Sum_{n>=1} a(n)x^n/n! exp(-n^2*x).

A196304 G.f.: x = Sum_{n>=1} a(n)x^n/(1 + n(n+1)/2*x)^n.

A229806 G.f.: Sum_{n>=1} a(n)x^n / (1 + nx)^(n^2) = x.

A275763 G.f.: x = Sum_{n>=1} a(n-1) * x^n / Product_{k=1..n} (1 + nkx).