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
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.
Programs
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GAP
List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/8^k))); # G. C. Greubel, Jan 29 2020
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Magma
[Denominator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // G. C. Greubel, Jan 29 2020
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Maple
seq(denom(add(binomial(2*k, k)/8^k, k = 0 .. n)), n = 0..25); # G. C. Greubel, Jan 29 2020
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Mathematica
Table[Denominator[Sum[Binomial[2*k, k]/8^k, {k,0,n}]], {n,0,25}] (* G. C. Greubel, Jan 29 2020 *) -
PARI
{a(n) = if( n<0, 0, 2^(3*n - subst( Pol( binary( n ) ), x, 1) ) ) } /* Michael Somos, Aug 23 2007 */ -
Sage
[denominator( binomial(1/4, n) ) for n in (0..25)] # G. C. Greubel, Jan 29 2020
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)
Comments