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A082159
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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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5
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1, 3, 39, 1206, 69189, 6416568, 881032059, 168514815360, 42934911510249, 14081311783382400, 5786296490491543599, 2914663547018935095552, 1767539279001227299807725, 1271059349855055258673975296, 1069996840045068513065229943875
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OFFSET
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0,2
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COMMENTS
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This is the first column of the array A082171.
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LINKS
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FORMULA
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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.
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MATHEMATICA
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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}];
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PROG
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(PARI) 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
(Magma)
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;
(SageMath)
@CachedFunction
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))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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