OFFSET
1,2
COMMENTS
a(n+1) is the moment of order n for the probability density function rho(x) = Pi^(-3/2)*sqrt(x/2)*exp(x/2)/(1-erf^2(i*sqrt(x/2))) on the interval 0..infinity, where erf is the error function and i=sqrt(-1). - Groux Roland, Nov 10 2009
REFERENCES
E. W. Bowen, Letter to N. J. A. Sloane, Aug 27 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..300
E. W. Bowen & N. J. A. Sloane, Correspondence, 1976
Colin K, Is it possible to make Mathematica reformulate an expression in a more numerically stable way?. see second answer by whuber.
FORMULA
a(n) = (1/2) * A000698(n+1), n > 0.
x + (5/2)*x^2 + (37/3)*x^3 + (353/4)*x^4 + (4081/5)*x^5 + (55205/6)*x^6 + ... = log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) where [1, 1, 3, 15, 105, 945, 10395, ...] = A001147(double factorials). - Philippe Deléham, Jun 20 2006
G.f.: ( 1/Q(0) - 1)/x where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: (2/x)/G(0) - 1/x, where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: 1/(2*x^2) - 1/(2*x) - G(0)/(2*x^2), where G(k) = 1 - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
L.g.f.: log(1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
MAPLE
df := proc(n) product(2*k-1, k=1..n) end: a[1] := 1: for n from 2 to 30 do a[n] := n*df(n)-sum(a[k]*df(n-k), k=1..n-1) od;
MATHEMATICA
CoefficientList[Series[D[Log[Sum[(2n-1)!!x^n, {n, 0, 17}]], x], {x, 0, 16}], x] [From Wouter Meeussen, Mar 21 2009]
a[ n_] := If[ n < 1, 0, n Coefficient[ Normal[ Series[ Log @ Erfc @ Sqrt @ x, {x, Infinity, n}] + x + Log[ Sqrt [Pi x]]] /. x -> -1 / 2 / x, x, n]] (* Michael Somos, May 28 2012 *)
PROG
(PARI) {a(n) = if( n<1, 0, n++; polcoeff( 1 - 1 / (2 * sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n))), n))} /* Michael Somos, May 28 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, following a suggestion from E. W. Bowen, Aug 27 1976
EXTENSIONS
Description corrected by Jeremy Magland (magland(AT)math.byu.edu), Jan 07 2000
More terms from Emeric Deutsch, Dec 21 2003
STATUS
approved