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%I #35 Sep 08 2022 08:45:12
%S 1,4,32,128,2048,8192,65536,262144,8388608,33554432,268435456,
%T 1073741824,17179869184,68719476736,549755813888,2199023255552,
%U 140737488355328,562949953421312,4503599627370496,18014398509481984
%N Denominators of the coefficients of powers of n^(-1) in the Romanovsky series expansion of the mean of the standard deviation from a normal population.
%C Is this the same sequence as A123854? - _N. J. A. Sloane_, Mar 21 2007
%C Almost certainly this is the same as A123854. - _Michael Somos_, Aug 23 2007
%C Asymptotic expansion of Gamma(N/2) / Gamma((N-1)/2) = (N/2)^(1/2) * (c(0) + c(1)/N + c(2)/N^2 + ... ). a(n) = denominator(c(n)). - _Michael Somos_, Aug 23 2007
%D V. Romanovsky, On the Moments of the Standard Deviation and of the Correlation Coefficient in Samples from Normal, Metron 5(4) (1925), 3-46.
%H G. C. Greubel, <a href="/A088802/b088802.txt">Table of n, a(n) for n = 0..500</a>
%H F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://arxiv.org/abs/1009.4274">Lehmer's Interesting Series</a>, arXiv:1009.4274 [math-ph], 2010-2011. See the unnumbered table on p. 7.
%H F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.116">Lehmer's interesting series</a>, Amer. Math. Monthly, 120 (2013), 116-130. See Table 4.
%H D. H. Lehmer, <a href="https://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Amer. Math. Monthly, 92(7) (1985), 449-457.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StandardDeviationDistribution.html">Standard Deviation Distribution</a>.
%F From _G. C. Greubel_, Jan 29 2020: (Start)
%F a(n) = denominator(Sum_{k=0..n} binomial(2*k, k)/8^k).
%F a(n) = denominator(binomial(1/4, n)). (End)
%p seq(denom(add(binomial(2*k, k)/8^k, k = 0 .. n)), n = 0..25); # _G. C. Greubel_, Jan 29 2020
%t Table[Denominator[Sum[Binomial[2*k, k]/8^k, {k,0,n}]], {n,0,25}] (* _G. C. Greubel_, Jan 29 2020 *)
%o (PARI) {a(n) = if( n<0, 0, 2^(3*n - subst( Pol( binary( n ) ), x, 1) ) ) } /* _Michael Somos_, Aug 23 2007 */
%o (Magma) [Denominator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // _G. C. Greubel_, Jan 29 2020
%o (Sage) [denominator( binomial(1/4, n) ) for n in (0..25)] # _G. C. Greubel_, Jan 29 2020
%o (GAP) List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/8^k))); # _G. C. Greubel_, Jan 29 2020
%Y Cf. A088801, A126963, A143503.
%K nonn,frac
%O 0,2
%A _Eric W. Weisstein_, Oct 16 2003
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