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

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

0,5

Comments

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

Links

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.

Formula

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.

From Seiichi Manyama, Apr 11 2026: (Start)

log(1+k^3*x) = Sum_{j>=1} T(j,k)/j * (x/(1 + (j+k)^3*x))^j.

1 = Sum_{j>=0} T(j,k) * binomial(j+m-1,j) * x^j/(1 + (j+k)^3*x)^(j+m) for m >= 1.

1 = Sum_{j>=0} T(j,k) * x^j/j! * exp(-(j+k)^3*x). (End)

Example

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;

Mathematica

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 *)

Prog

(Magma)
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
(SageMath)
@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

Crossrefs

Columns k=1..3 give A082158, A395085, A395086.

Cf. A082158, A082162, A082169.

Sequence in context: A383684 A331814 A387124 * A136377 A103706 A134990

Adjacent sequences: A082167 A082168 A082169 * A082171 A082172 A082173

Keyword

easy,nonn,tabl

Author

Valery A. Liskovets, Apr 09 2003

Status

approved