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A082687 Numerator of Sum_{k=1..n} 1/(n+k). 11
1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 7751493599, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

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

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

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

Seems to be the denominator of the harmonic mean of the first n hexagonal numbers. - Colin Barker, Nov 19 2014

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Hilbert Matrix.

FORMULA

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

Numerator of Psi(2*n+1) - Psi(n+1). - Vladeta Jovovic, Aug 24 2003

a(n) = numerator(Sum_{k=1..2*n} 1/k) - Sum_{k=1..n} 1/k). - Alexander Adamchuk, Apr 11 2006

a(n) = numerator(Sum_{j=1..n} (Sum_{i=1..n} 1/(i+j-1))). - Alexander Adamchuk, Apr 11 2006

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)).

EXAMPLE

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.

n=2: HilbertMatrix(n,n)

1 1/2

1/2 1/3

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

The n X n Hilbert matrix begins:

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

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

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

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

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

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

MAPLE

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

seq(a(n), n=1..20); # Peter Luschny, Nov 02 2017

MATHEMATICA

Numerator[Sum[1/k, {k, 1, 2*n}] - Sum[1/k, {k, 1, n}]] (* Alexander Adamchuk, Apr 11 2006 *)

Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] (* Alexander Adamchuk, Apr 11 2006 *)

Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)

PROG

(PARI) a(n) = numerator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

CROSSREFS

Bisection of A058313. A082688 (denominators).

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

Sequence in context: A093168 A276231 A097493 * A117731 A155010 A292807

Adjacent sequences:  A082684 A082685 A082686 * A082688 A082689 A082690

KEYWORD

frac,nonn,easy

AUTHOR

Benoit Cloitre, Apr 12 2003

STATUS

approved

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)