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A085614
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Number of elementary arches of size n.
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5
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1, 3, 16, 105, 768, 6006, 49152, 415701, 3604480, 31870410, 286261248, 2604681690, 23957864448, 222399744300, 2080911654912, 19604537460045, 185813170126848, 1770558814528770, 16951376923852800, 162984598242674670
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OFFSET
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1,2
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LINKS
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Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.
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FORMULA
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G.f. is the series reversion of x-3*x^2+2*x^3.
a(n) = 2^n*(3*n)!!/((n+1)!*n!!). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), May 25 2007
G.f.: 1/6*sqrt(3)*sin(1/3*arcsin(6*sqrt(3)*x))-1/2*cos(1/3*arcsin(6*sqrt(3)*x)). - Vaclav Kotesovec, Oct 21 2012
Conjecture: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) -12*(3*n-5)*(3*n-7)*a(n-2) -12*(3*n-8)*(3*n-10)*a(n-3) = 0. - R. J. Mathar, Oct 18 2013
a(n) ~ 2^(n - 3/2) * 3^(3*n/2 - 1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2017
a(n) = 4^n Gamma((3*n + 2)/2)/(Gamma((n + 2)/2)*(n + 1)!).
a(n) = (4^n*((n + 2)/2)_n)/(n + 1)!, where (x)_k is the Pochhammer symbol. (End)
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MAPLE
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with(combstruct); ar := {EA = Union(Sequence(EA, card >= 2), Prod(Z, Sequence(EA), Sequence(EA))), C=Union(Z, Prod(Z, Z, Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA, card>=1), Prod(Z, Sequence(EA), Sequence(EA))))))}; seq(count([EA, ar], size=i), i=1..20);
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MATHEMATICA
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Rest[CoefficientList[Series[1/6*Sqrt[3]*Sin[1/3*ArcSin[6*Sqrt[3]*x]] - 1/2*Cos[1/3*ArcSin[6*Sqrt[3]*x]], {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 21 2012 *)
Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + 2*x^3, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Aug 22 2017 *)
Table[4^n Gamma[(3n + 2)/2]/(Gamma[(n + 2)/2](n + 1)!), {n, 0, 20}]
Table[4^n Pochhammer[(n + 2)/2, n]/(n + 1)!, {n, 0, 20}] (* End *)
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PROG
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(PARI) a(n)=if(n<1, 0, polcoeff(serreverse(x-3*x^2+2*x^3+x*O(x^n)), n))
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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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