A004208 a(n) = n * (2*n - 1)!! - Sum_{k=0..n-1} a(k) * (2*n - 2*k - 1)!!.

1, 5, 37, 353, 4081, 55205, 854197, 14876033, 288018721, 6138913925, 142882295557, 3606682364513, 98158402127761, 2865624738913445, 89338394736560917, 2962542872271918593, 104128401379446177601, 3867079042971339087365, 151312533647578564021477

Offset: 1

Author

N. J. A. Sloane, following a suggestion from E. W. Bowen, Aug 27 1976

Keywords

• nonn

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.

Crossrefs

Cf. A000698.

Programs

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,0,19}]],x],{x,0,18}],x] (* 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 *)

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 */

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

Extensions

Description corrected by Jeremy Magland (magland(AT)math.byu.edu), Jan 07 2000

More terms from Emeric Deutsch, Dec 21 2003