1,2

Numerator of sum{k=0..n, 1/((k+1)(2k+1))} (denominator is A111876). - Paul Barry (pbarry(AT)wit.ie), 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 (alex(AT)kolmogorov.com), 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 (1/k, k=1..n) is n-th Harmonic Number. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

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 (alex(AT)kolmogorov.com), Jul 16 2006

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 (qq-quet(AT)mindspring.com), May 20 2007

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

limit n-->infinity sum(k=1, n, 1/(n+k))=log(2)

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

a(n) = Numerator[Sum[1/k,{k,1,2*n}] - Sum[1/k,{k,1,n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

a(n) = Numerator[Sum[Sum[1/(i+j-1), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

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 ...

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

Table[Numerator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 20}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

Cf. A058313, A082688 (denominators).

Bisection of A058313.

Cf. A001008, A002805, A058313, A058312.

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

Cf. A117731.

Sequence in context: A075996 A093168 A097493 this_sequence A117731 A080174 A127729

Adjacent sequences: A082684 A082685 A082686 this_sequence A082688 A082689 A082690

frac,nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 12 2003