%I #11 May 24 2015 19:23:41

%S 1,-3,-7,-9,59,483,-2323,-42801,923923,30055311,-170042041,

%T -8639161167,99976667055,7336972779615,-42962450319915,

%U -4309733345367105,203289825295660035,26751125064470578695,-158415664732997134045,-26488943422458070446915

%N Numerators of coefficients of powers of n^(-1) in the Romanovsky series expansion of the mean of the standard deviation from a normal population.

%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) = numerator(c(n)). - _Michael Somos_, Aug 23 2007

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StandardDeviationDistribution.html">Standard Deviation Distribution</a>

%e b(N) = 1 - 3/(4N) - 7/(32N^2) - 9/(128N^3) + ...

%t a[ n_] := If[ n < 0, 0, Module[{A = 1}, Do[ A += x^k / (4 k) SeriesCoefficient[ (A /. x -> x / (1 + 2 x))^2 - (A/(1 - x))^2 / (1 + 2 x) + O[x]^(k + 2), k + 1], {k, n}]; Numerator@Coefficient[A, x, n]]]; (* _Michael Somos_, May 24 2015 *)

%o (PARI) {a(n) = my(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x^2 * O(x^k); A += x^k/4/k * polcoeff( subst( A, x, x/(1+2*x))^2 - A^2/(1-x)^2/(1+2*x), k+1 ) ); numerator( polcoeff( A, n ) ) ) }; /* _Michael Somos_, Aug 23 2007 */

%Y Cf. A088802.

%K sign,frac

%O 0,2

%A _Eric W. Weisstein_, Oct 16 2003

%E a