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A088801
Numerators of coefficients of powers of n^(-1) in the Romanovsky series expansion of the mean of the standard deviation from a normal population.
1
1, -3, -7, -9, 59, 483, -2323, -42801, 923923, 30055311, -170042041, -8639161167, 99976667055, 7336972779615, -42962450319915, -4309733345367105, 203289825295660035, 26751125064470578695, -158415664732997134045, -26488943422458070446915
OFFSET
0,2
COMMENTS
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
LINKS
Eric Weisstein's World of Mathematics, Standard Deviation Distribution
EXAMPLE
b(N) = 1 - 3/(4N) - 7/(32N^2) - 9/(128N^3) + ...
MATHEMATICA
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 *)
PROG
(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 */
CROSSREFS
Cf. A088802.
Sequence in context: A074339 A355732 A115164 * A003033 A193945 A087147
KEYWORD
sign,frac
AUTHOR
Eric W. Weisstein, Oct 16 2003
EXTENSIONS
a
STATUS
approved