More considered contemplation of the evidence suggests I had over-egged the reliability of these extrapolations, a safer estimate being merely b = 0.13 ... Also this now seems unlikely to be capable of further refinement, at any rate via my customary Aitken delta^2 or Wynn rho. WFL On 3/15/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Experimental evidence suggests that in fact strictly
f(n) ~ b*c^(n^2)
where c = sqrt(27/16) is Greathouse's constant, and b = 0.12018 ... If prevailed upon to do so, I could doubtless winkle out more digits of b ; but prefer to wait to see if anybody (Charles? Bill?) is prepared to come up with an exact value first!
Fred Lunnon
On Tue, Mar 11, 2014 at 1:03 PM, 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
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