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%I #67 Jul 24 2023 02:38:21
%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
%C From _Peter Bala_, Feb 16 2022: (Start)
%C 2*Sum_{k = 1..n} 1/(n+k) = 1 + 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + - .... Cf. A101028.
%C 2*Sum_{k = 1..n} 1/(n+k) = n - (1 + 1/2^2)*n*(n-1)/(n+1) + (1/2^2 + 1/3^2)*n*(n-1)*(n-2)/((n+1)*(n+2)) - (1/3^2 + 1/4^2)*n*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + (1/4^2 + 1/5^2)*n*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - + .... Cf. A007406 and A120778.
%C These identities allow us to extend the definition of Sum_{k = 1..n} 1/(n+k) to non-integral values of n. (End)
%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 -> oo} 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
%o (Magma) [Numerator((HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // _G. C. Greubel_, Jul 24 2023
%o (SageMath) [numerator(harmonic_number(2*n,1) - harmonic_number(n,1)) for n in range(1,41)] # _G. C. Greubel_, Jul 24 2023
%Y Bisection of A058313, A082688 (denominators).
%Y Cf. A001008, A002805, A005249, A007406, A058312.
%Y Cf. A086881, A098118, A101028, A117731, A120778.
%K frac,nonn,easy
%O 1,2
%A _Benoit Cloitre_, Apr 12 2003
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