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),...

The first column is A082158.

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.

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

Cf. A082158, A082162, A082169.

Sequence in context: A253287 A357514 A331814 * A136377 A103706 A134990

Adjacent sequences: A082167 A082168 A082169 * A082171 A082172 A082173

Keyword

easy,nonn,tabl

Author

Valery A. Liskovets, Apr 09 2003

Status

approved