|
|
A082169
|
|
Deterministic completely defined quasi-acyclic automata with 2 inputs, n transient and k absorbing labeled states.
|
|
6
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
Array read by antidiagonals: (0,1),(0,2),(1,1),(0,3),...
|
|
LINKS
|
|
|
FORMULA
|
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.
|
|
EXAMPLE
|
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;
|
|
MATHEMATICA
|
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}];
|
|
PROG
|
(Magma)
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) >;
(SageMath)
@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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|