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, -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

Table of n, a(n) for n=0..19.

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

Adjacent sequences: A088798 A088799 A088800 * A088802 A088803 A088804

Keyword

sign,frac

Author

Eric W. Weisstein, Oct 16 2003

Extensions

a

Status

approved