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.

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

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: A338322 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