login
A088802
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.
12
1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984
OFFSET
0,2
COMMENTS
Is this the same sequence as A123854? - N. J. A. Sloane, Mar 21 2007
Almost certainly this is the same as A123854. - Michael Somos, Aug 23 2007
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
REFERENCES
V. Romanovsky, On the Moments of the Standard Deviation and of the Correlation Coefficient in Samples from Normal, Metron 5(4) (1925), 3-46.
LINKS
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. See the unnumbered table on p. 7.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. See Table 4.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Eric Weisstein's World of Mathematics, Standard Deviation Distribution.
FORMULA
From G. C. Greubel, Jan 29 2020: (Start)
a(n) = denominator(Sum_{k=0..n} binomial(2*k, k)/8^k).
a(n) = denominator(binomial(1/4, n)). (End)
MAPLE
seq(denom(add(binomial(2*k, k)/8^k, k = 0 .. n)), n = 0..25); # G. C. Greubel, Jan 29 2020
MATHEMATICA
Table[Denominator[Sum[Binomial[2*k, k]/8^k, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Jan 29 2020 *)
PROG
(PARI) {a(n) = if( n<0, 0, 2^(3*n - subst( Pol( binary( n ) ), x, 1) ) ) } /* Michael Somos, Aug 23 2007 */
(Magma) [Denominator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // G. C. Greubel, Jan 29 2020
(Sage) [denominator( binomial(1/4, n) ) for n in (0..25)] # G. C. Greubel, Jan 29 2020
(GAP) List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/8^k))); # G. C. Greubel, Jan 29 2020
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Oct 16 2003
STATUS
approved