A004041 Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).

1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500

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