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 A082687 Numerator of Sum_{k=1..n} 1/(n+k). 11

%I

%S 1,7,37,533,1627,18107,237371,95549,1632341,155685007,156188887,

%T 3602044091,18051406831,7751493599,225175759291,13981692518567,

%U 14000078506967,98115155543129,3634060848592973,3637485804655193

%N Numerator of Sum_{k=1..n} 1/(n+k).

%C Numerator of Sum_{k=0..n-1} 1/((k+1)(2k+1)) (denominator is A111876). - _Paul Barry_, Aug 19 2005

%C Numerator of the sum of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - _Alexander Adamchuk_, Apr 11 2006

%C Numerator of the 2n-th alternating harmonic number H'(2n) = Sum ((-1)^(k+1)/k, k=1..2n). H'(2n) = H(2n) - H(n), where H(n) = Sum_{k=1..n} 1/k is the n-th Harmonic Number. - _Alexander Adamchuk_, Apr 11 2006

%C a(n) almost always equals A117731(n) = numerator(n*Sum_{k=1..n} 1/(n+k)) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)) but differs for n = 14, 53, 98, 105, 111, 114, 119, 164. - _Alexander Adamchuk_, Jul 16 2006

%C Sum_{k=1..n} 1/(n+k) = n!^2 *Sum_{j=1..n} (-1)^(j+1) /((n+j)!(n-j)!j). - _Leroy Quet_, May 20 2007

%C Seems to be the denominator of the harmonic mean of the first n hexagonal numbers. - _Colin Barker_, Nov 19 2014

%C Numerator of 2*n*binomial(2*n,n)*Sum_{k = 0..n-1} (-1)^k* binomial(n-1,k)/(n+k+1)^2. Cf. A049281. - _Peter Bala_, Feb 21 2017

%H T. D. Noe, <a href="/A082687/b082687.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.

%F limit n-->infinity Sum_{k=1..n} 1/(n+k) = log(2).

%F Numerator of Psi(2*n+1) - Psi(n+1). - _Vladeta Jovovic_, Aug 24 2003

%F a(n) = numerator(Sum_{k=1..2*n} 1/k) - Sum_{k=1..n} 1/k). - _Alexander Adamchuk_, Apr 11 2006

%F a(n) = numerator(Sum_{j=1..n} (Sum_{i=1..n} 1/(i+j-1))). - _Alexander Adamchuk_, Apr 11 2006

%F The o.g.f for Sum_{k=1..n} 1/(n+k) is f(x) = (sqrt(x)*log((1+sqrt(x))/(1-sqrt(x))) + log(1-x))/(2*x*(1-x)).

%e H'(2n) = H(2n) - H(n) = {1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, 95549/144144, 1632341/2450448, 155685007/232792560, ...}, where H(n) = A001008/A002805.

%e n=2: HilbertMatrix(n,n)

%e 1 1/2

%e 1/2 1/3

%e so a(2) = Numerator(1 + 1/2 + 1/2 + 1/3) = Numerator(7/3) = 7.

%e The n X n Hilbert matrix begins:

%e 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...

%e 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...

%e 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...

%e 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...

%e 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...

%e 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

%p a := n -> numer(harmonic(2*n) - harmonic(n)):

%p seq(a(n), n=1..20); # _Peter Luschny_, Nov 02 2017

%t Numerator[Sum[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] (* _Alexander Adamchuk_, Apr 11 2006 *)

%t Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] (* _Alexander Adamchuk_, Apr 11 2006 *)

%t Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Numerator (* _Eric W. Weisstein_, Dec 14 2017 *)

%o (PARI) a(n) = numerator(sum(k=1, n, 1/(n+k))); \\ _Michel Marcus_, Dec 14 2017

%Y Bisection of A058313. A082688 (denominators).

%Y Cf. A001008, A002805, A058312, A098118, A086881, A005249, A001008, A117731.

%K frac,nonn,easy

%O 1,2

%A _Benoit Cloitre_, Apr 12 2003

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Last modified March 19 00:55 EDT 2018. Contains 300772 sequences. (Running on oeis4.)