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A054745 Number of nonisomorphic binary n-state automata without output under input permutations. 10
1, 1, 7, 74, 1474, 41876, 1540696, 68343112, 3540691525, 209612916303, 13957423192794, 1032436318269648, 83993175608894096, 7453446303042245261, 716451740543945788671, 74159075140708644544128, 8223831291824019614386868, 972718473204236819072891710 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also isomorphism classes of unordered pairs of endofunctions i.e. an unorder pair {f,g} of functions from {1,...,n} to itself. - Christian G. Bower, Dec 18 2003

REFERENCES

F. Harary and E. Palmer, Graphical Enumeration, 1973.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..45

M. A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1, 1965, p. 110.

FORMULA

a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2=2} (fix A[s_1, s_2, ...;t_1, t_2]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!)) where fix A[...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*t_j)). - Christian G. Bower, Dec 18 2003

MAPLE

with(numtheory):

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},

      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))

    end:

a:= proc(n) option remember; add(add(mul(mul(add(coeff(s, x, d)

      *d, d=divisors(ilcm(i, j)))^(igcd(i, j)*coeff(s, x, i)*

      coeff(t, x, j)), j=1..degree(t)), i=1..degree(s))

      /mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t))

      /mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))

      , t=b(2$2)), s=b(n$2))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2014

MATHEMATICA

Unprotect[Power]; 0^0 = 1; b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Table[ Map[ Function[{p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}] // Flatten // Union]] ; a[n_] := a[n] = Sum[ Sum[ Product[ Product[ Sum[ Coefficient[s, x, d] *d, {d, Divisors[LCM[i, j]]}]^(GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j]), {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i] * Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}] / Product[i^Coefficient[s, x, i] * Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] , {t, b[2, 2]}], {s, b[n, n]}] ; Table[a[n], {n, 0, 20}](* Jean-Fran├žois Alcover, Mar 16 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A001372, A054050, A054732, A054746.

Sequence in context: A266305 A098118 A097821 * A197091 A174243 A157706

Adjacent sequences:  A054742 A054743 A054744 * A054746 A054747 A054748

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Apr 22 2000

EXTENSIONS

More terms from Alois P. Heinz, Aug 15 2014

STATUS

approved

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Last modified August 15 13:13 EDT 2018. Contains 313764 sequences. (Running on oeis4.)