login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004041 Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)). 18
1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

n-th elementary symmetric function of the first n+1 odd positive integers.

Also the determinant of the n X n matrix given by m(i,j) = 2*i + 2 = if i = j and otherwise 1. For example, Det[{{4, 1, 1, 1, 1, 1}, {1, 6, 1, 1, 1, 1}, {1, 1, 8, 1, 1, 1}, {1, 1, 1, 10, 1, 1}, {1, 1, 1, 1, 12, 1}, {1, 1, 1, 1, 1, 14}}] = 264207 = a(6). - John M. Campbell, May 20 2011

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..403

J. Courtiel, K. Yeats, Terminal chords in connected chord diagrams, arXiv:1603.08596 [math.CO], 2016; e.g.f. in Remark 1 B_1(z).

FORMULA

a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).

a(n) is coefficient of x^(2*n+2) in (arctanh x)^2, multiplied by (n + 1)*(2*n + 1)!!.

a(n) = Sum_{i=k+1..n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 1, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001

a(n) ~ 2^(1/2)*log(n)*n*(2n/e)^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

E.g.f.: 1/2*(1 - 2*x)^(-3/2)*(2 - log(1 - 2*x)). - Vladeta Jovovic, Feb 19 2003

Sum_{n>=1} a(n-1)/(n!*n*2^n)) = (Pi/2)^2. - Philippe Deléham, Aug 12 2003

For n >= 1, a(n-1) = 2^(n-1)*n!*(Sum_{k=0..n-1} (-1)^k*binomial(1/2, k)/(n - k)). - Milan Janjic, Dec 14 2008

Recurrence: a(n) = 4*n*a(n-1) - (2*n - 1)^2*a(n-2). - Vladimir Reshetnikov, Oct 13 2016

EXAMPLE

(arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...

MATHEMATICA

Table[(-1)^(n + 1)* Sum[(-2)^(n - k) k (-1)^(n - k) StirlingS1[n + 1, k + 1], {k, 0, n}], {n, 1, 18}] (* Zerinvary Lajos, Jul 08 2009 *)

FunctionExpand@Table[(2 n + 1)!! (Log[4] + HarmonicNumber[n + 1/2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 13 2016 *)

CROSSREFS

Cf. A000254, A024199, A049034.

Cf. A002428.

From Johannes W. Meijer, Jun 08 2009: (Start)

Equals second left hand column of A028338 triangle.

Equals second right hand column of A109692 triangle.

Equals second left hand column of A161198 triangle divided by 2.

(End)

Sequence in context: A025550 A350669 A067545 * A220353 A089465 A220214

Adjacent sequences:  A004038 A004039 A004040 * A004042 A004043 A004044

KEYWORD

nonn

AUTHOR

Joe Keane (jgk(AT)jgk.org)

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 27 14:22 EDT 2022. Contains 357062 sequences. (Running on oeis4.)