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%I #18 Nov 07 2023 04:32:55
%S 1,4,38,587,12607,347158,11668113,463118041,21199488803,1099465138203,
%T 63715991036964,4080500855334901,286178278238641752,
%U 21813909692571410084,1795659553423061982001,158754024731440581761116,15002712207593790179795284,1509215071938528737864389367,161017605699030302902310357883
%N Number of labeled forests of rooted Greg hypertrees with n white vertices.
%C A Greg hypertree is a hypertree with black and white vertices, such that black vertices are unlabeled and have at least two incoming edges.
%F E.g.f: series reversion of (log(1+t)-exp(t)+t+1)*exp(-t).
%F a(n) ~ sqrt((1+s)*(2+s)/((1+r)*(3 + s*(3+s)))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where s = 0.3900539630495916058133890253422601894372373496844... is the root of the equation exp(-s + 1/(1+s)) = 1+s and r = exp(-s)*(1 + 1/(1+s)) - 1 = 0.1640664235584946357534702598223332293549130374395... - _Vaclav Kotesovec_, Oct 24 2023
%t Rest[CoefficientList[InverseSeries[Series[E^-x (1 + x + Log[1 + x]) - 1, {x, 0, 20}], x], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 24 2023 *)
%o (PARI) my(t='t+O('t^25)); Vec(serlaplace(serreverse((log(1+t)-exp(t)+t+1)*exp(-t)))) \\ _Michel Marcus_, Oct 21 2023
%Y Cf. A005264, A052888.
%K nonn,easy
%O 1,2
%A _Paul Laubie_, Oct 21 2023