|
| |
|
|
A004041
|
|
Scaled sums of odd reciprocals: a(n) = (2*n+1)!! * sum(k=0..n, 1/(2*k+1) ).
|
|
11
| |
|
|
1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500
(list; graph; refs; listen; history; 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
|
|
|
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)!!.
sum[(-1)^(k+1-i) 2^(n-1) binomial(i-1, k) s1(n, i), i=k+1..n] 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*2^n*e^-n*n^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 (vladeta(AT)eunet.rs), Feb 19 2003
Sum(n>=1; a(n-1)/(n!*n*2^n)) = (Pi/2)^2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 12 2003
For n>=1, a(n-1)=2^(n-1)*n!*sum((-1)^k*binomial(1/2,k)/(n-k),k=0..n-1); [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]
|
|
|
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}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
|
|
|
CROSSREFS
| Cf. A000254, A024199, A049034.
Cf. A002428.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), 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: A141763 A025550 A067545 * A089465 A106174 A056814
Adjacent sequences: A004038 A004039 A004040 * A004042 A004043 A004044
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Joe Keane (jgk(AT)jgk.org)
|
| |
|
|
