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A121355
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Number of transitive PSL_2(ZZ) actions on a finite labeled set of size n.
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5
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1, 1, 8, 48, 120, 2640, 30240, 201600, 4838400, 96163200, 1037836800, 30496435200, 828193766400, 13686991718400, 450537408921600, 15880397524992000, 356398802952192000, 13410127414075392000, 569542360114151424000, 16614774394239909888000
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OFFSET
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1,3
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COMMENTS
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Equivalently, the number of different connected labeled trivalent diagrams of size n.
Also the number of (r,s) pair of permutations in S_n, which generate a transitive action and 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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FORMULA
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If A(z) = g.f. of a(n) and B(z) = g.f. of A121357 then A(z) = log(B(z)).
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MAPLE
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N := 100 : exs2:=sort(convert(taylor(exp(t+t^2/2), t, N+1), polynom), t, ascending) : exs3:=sort(convert(taylor(exp(t+t^3/3), t, N+1), polynom), t, ascending) : exs23:=sort(add(op(n+1, exs2)*op(n+1, exs3)/(t^n/ n!), n=0..N), t, ascending) : logexs23:=sort(convert(taylor(log(exs23), t, N+1), polynom), t, ascending) : sort(add(op(n, logexs23)*n!, n=1..N), t, ascending);
# Alternatively:
A121355_list := proc(len) local s, p; s := f -> seq(n!*coeff(series(f, z, n+1), z, n), n=0..len); p := m -> s(exp(z+z^m/m)); s(log(add((p(2)[n+1]*p(3)[n+1])*z^n/n!, n=0..len))) end: # Peter Luschny, Nov 16 2015
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MATHEMATICA
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m = 19; exs2 = Series[Exp[t + t^2/2], {t, 0, m + 1}] // Normal; exs3 = Series[Exp[t + t^3/3], {t, 0, m + 1}] // Normal; exs23 = Sum[exs2[[n + 1]]*exs3[[n + 1]]/(t^n/n!), {n, 0, m}]; logexs23 = Series[Log[exs23], {t, 0, m}] // Normal; CoefficientList[logexs23, t]*Range[0, m]! // Rest (* Jean-François Alcover, Sep 06 2013, translated and adapted from Samuel Vidal's Maple program *)
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PROG
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(PARI) N=20; x='x+O('x^(N+1));
A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
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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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