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A121352
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Number of different, not necessarily connected, unlabeled trivalent diagrams of size n.
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8
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1, 1, 2, 4, 7, 10, 24, 37, 63, 112, 200, 318, 607, 1058, 1814, 3247, 6004, 10316, 19048, 35478, 63496, 117023, 223822, 408121, 766661, 1484363, 2775201, 5270079, 10357605, 19714259, 37970066, 75439670, 146103241, 284719527, 571706625
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OFFSET
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0,3
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COMMENTS
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Equivalently, the number of isomorphism class of PSL_2(ZZ) actions on finite sets of size n.
Also the number of (r,s) pair of permutations up to simultaneous conjugation, in S_n for which r is involutive i.e. r^2 = id and s is of weak order three i.e. s^3 = id.
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LINKS
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MAPLE
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mu := k -> `if`( k mod 2 = 0, 2/k, 1/k ) : nu := k -> `if`( k mod 3 = 0, 3/k, 1/k ) : u := (k, n) -> add(mu(k)^(n-2*k2)/(n-2*k2)!/k2!/(2*k)^k2, k2=0..floor(n/ 2)) : v := (k, n) -> add(nu(k)^(n-3*k3)/(n-3*k3)!/k3!/(3*k)^k3, k3=0..floor(n/ 3)) : N := 100 : ZF := 1 : for k from N to 1 by -1 do ZF := rem(ZF * add(n!*k^n*u(k, n)*v(k, n)*t^(k*n), n = 0..floor(N/ k)), t^(N+1), t) ; end do : sort(ZF, t, ascending);
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MATHEMATICA
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max = 34; mu[k_] := If[Mod[k, 2] == 0, 2/k, 1/k]; nu[k_] := If[Mod[k, 3] == 0, 3/k, 1/k]; u[k_, n_] := Sum[ mu[k]^(n - 2*k2) / (((n - 2*k2)!*k2!)*(2*k)^k2), {k2, 0, Floor[n/2]}]; v[k_, n_] := Sum[ nu[k]^(n - 3*k3) / (((n - 3*k3)!*k3!)*(3*k)^k3), {k3, 0, Floor[n/3]}]; ZF = 1; For[k = max, k >= 1, k--, ZF = PolynomialMod[ ZF*Sum[ n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}], t^(max + 1)]]; CoefficientList[ZF, t](* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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