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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. 8
1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984 (list; graph; refs; listen; history; text; internal format)
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

G. C. Greubel, Table of n, a(n) for n = 0..500

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

Cf. A088801, A126963, A143503.

Sequence in context: A267668 A239056 A088658 * A123854 A301843 A332430

Adjacent sequences:  A088799 A088800 A088801 * A088803 A088804 A088805

KEYWORD

nonn,frac

AUTHOR

Eric W. Weisstein, Oct 16 2003

STATUS

approved

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Last modified August 5 16:03 EDT 2020. Contains 336213 sequences. (Running on oeis4.)