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%I #47 Feb 04 2022 02:16:10
%S 1,4,23,176,1689,19524,264207,4098240,71697105,1396704420,29985521895,
%T 703416314160,17901641997225,491250187505700,14459713484342175,
%U 454441401368236800,15188465029114325025,537928935889764226500
%N Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
%C n-th elementary symmetric function of the first n+1 odd positive integers.
%C 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
%H Seiichi Manyama, <a href="/A004041/b004041.txt">Table of n, a(n) for n = 0..403</a>
%H J. Courtiel, K. Yeats, <a href="http://arxiv.org/abs/1603.08596">Terminal chords in connected chord diagrams</a>, arXiv:1603.08596 [math.CO], 2016; e.g.f. in Remark 1 B_1(z).
%F a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
%F a(n) is coefficient of x^(2*n+2) in (arctanh x)^2, multiplied by (n + 1)*(2*n + 1)!!.
%F 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
%F a(n) ~ 2^(1/2)*log(n)*n*(2n/e)^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
%F E.g.f.: 1/2*(1 - 2*x)^(-3/2)*(2 - log(1 - 2*x)). - _Vladeta Jovovic_, Feb 19 2003
%F Sum_{n>=1} a(n-1)/(n!*n*2^n)) = (Pi/2)^2. - _Philippe Deléham_, Aug 12 2003
%F 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
%F Recurrence: a(n) = 4*n*a(n-1) - (2*n - 1)^2*a(n-2). - _Vladimir Reshetnikov_, Oct 13 2016
%e (arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
%t 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 *)
%t FunctionExpand@Table[(2 n + 1)!! (Log[4] + HarmonicNumber[n + 1/2])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 13 2016 *)
%Y Cf. A000254, A024199, A049034.
%Y Cf. A002428.
%Y From _Johannes W. Meijer_, Jun 08 2009: (Start)
%Y Equals second left hand column of A028338 triangle.
%Y Equals second right hand column of A109692 triangle.
%Y Equals second left hand column of A161198 triangle divided by 2.
%Y (End)
%K nonn
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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