This sequence inter alia was also just being discussed under the "Hardin 'nuff" thread. Charles Greathouse opined
I think f(n) ~ sqrt(27/16)^(n^2), which is not the Lieb constant.
--- rather simpler than RWG's expression! [ "~" is used loosely --- it strictly applies only in the context of log f(n) ]. Fred Lunnon On 3/11/14, Bill Gosper <billgosper@gmail.com> wrote:
Besides Warren's proposals, Dodgson's method is valuable theoretically, helping Rich prove my Somos addition formulæ http://arxiv.org/abs/math/0703470 and Zeilberger prove the famously challenging Alternating Sign Matrix Theorem (A005130), the "Stirling's formula" for which is
Out[264]= (2^(5/12 - 2*n^2)*3^(-(7/36) + (3*n^2)/2)* E^((1/3)*HoldForm[Derivative[1][Zeta][-1]])*Pi^(1/3))/ (n^(5/36)*Gamma[1/3]^(2/3))
(the HoldForm to fend off that vexatious Glaisher symbol).
In[265]:= Table[N[ReleaseHold[%]], {n, 9}] // InputForm
Out[265]//InputForm= {1.0063254118710128, 2.003523267231662, 7.0056223910285915, 42.01915917750558, 429.12582410098327, 7437.518404899576, 218380.8077275304, 1.085146545456063*^7, 9.119184824937415*^8}
the actual sequence being
1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700,
Adamchik may have already worked this out someplace, in spite of the painful obstacles thrown up by Mathematica.
OEIS could probably use more such asymptotic formulæ. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun