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A097629
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a(n) = 2*(2n)^(n-2).
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8
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1, 2, 12, 128, 2000, 41472, 1075648, 33554432, 1224440064, 51200000000, 2414538435584, 126806761930752, 7340688973975552, 464436530178424832, 31886460000000000000, 2361183241434822606848, 187591757103747287810048
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OFFSET
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1,2
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COMMENTS
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Number of all unrooted directed trees on n nodes.
Ditrees are well-colored directed trees. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.
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LINKS
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FORMULA
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E.g.f.: A(x) = B(x)-B(x)^2, B(x) = e.g.f. of A052746 or A(x) = C(2*x)/2, C(x) = e.g.f. of A000272.
E.g.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} x^(2*n-1)/(2*n-1)! * A(x)^((4*n-1)/2) when offset=0: A(x) = Sum_{n>=0} a(n)*x^n/n!. - Paul D. Hanna, Sep 07 2012
E.g.f. satisfies: A(x) = 1/A(-x*A(x)^2) when offset=0. - Paul D. Hanna, Sep 07 2012
a(n) = sum(k=0..n-1, k!*stirling2(n-1,k)*binomial(2*n,k))/n. - Vladimir Kruchinin, Nov 19 2014
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MATHEMATICA
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Table[2*(2*n)^(n - 2), {n, 1, 50}] (* or *) With[{nmax = 40}, CoefficientList[Series[-LambertW[-2*x]*(1+LambertW[-2*x]/2)/2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 15 2017 *)
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PROG
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(PARI) /* E.g.f. when offset=0 satisfies: */
{a(n)=local(A=1+2*x); for(i=1, 21, A=1+2*sum(n=1, 21, x^(2*n-1)/(2*n-1)!*A^((4*n-1)/2))+x*O(x^n)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Sep 07 2012
for(n=0, 20, print1(a(n), ", "))
(Maxima) a(n):=sum(k!*stirling2(n-1, k)*binomial(2*n, k), k, 0, n-1)/(n); /* Vladimir Kruchinin, Nov 19 2014 */
(PARI) x='x+O('x^50); Vec(serlaplace(-lambertw(-2*x)*(1 + lambertw(-2*x)/2)/2)) \\ G. C. Greubel, Nov 15 2017
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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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