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%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/((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 (1/k, k=1..n) is n-th Harmonic Number. - _Alexander Adamchuk_, Apr 11 2006
%C a(n) almost always equals A117731[n] = Numerator[n*Sum[1/(n+k),{k,1,n}]] = Numerator[Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]], but differs for n=14,53,98,105,111,114,119,164.. - _Alexander Adamchuk_, Jul 16 2006
%C Sum{k=1 to n} 1/(n+k) = n!^2 *sum{j=1 to n} (-1)^(j+1) /((n+j)!(n-j)!j). - Leroy Quet May 20 2007
%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[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] - _Alexander Adamchuk_, Apr 11 2006
%F a(n) = Numerator[Sum[Sum[1/(i+j-1), {i, 1, n}], {j, 1, n}]] - _Alexander Adamchuk_, Apr 11 2006
%F The o.g.f for sum(k=1,n,1/(n+k))is
%F f(x)=(sqrt(x)*ln((1+sqrt(x))/(1-sqrt(x)))+ln(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 ...
%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
%Y Cf. A058313, A082688 (denominators).
%Y Bisection of A058313.
%Y Cf. A001008, A002805, A058313, A058312, A098118, A086881, A005249, A001008, A002805, A117731.
%K frac,nonn
%O 1,2
%A _Benoit Cloitre_, Apr 12 2003
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