A004208 a(n) = n * (2*n - 1)!! - Sum_{k=0..n-1} a(k) * (2*n - 2*k - 1)!!. (Formerly M3985)

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

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

Cf. A000698.

Sequence in context: A197713 A343993 A356848 * A198077 A208813 A112698

Adjacent sequences: A004205 A004206 A004207 * A004209 A004210 A004211

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