1,2

The numerators are given in A101028.

One third of the denominator of the finite differences of the series of sums 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

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

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

a(n)=denominator(s(n)) with s(n)=3*sum(1/((2*k-1)*k*(2*k+1)), k=1..n). See A101028 for more information.

a(n) = 1/3*Denominator[Sum[Sum[1/(i+j-1),{i,1,n+1}],{j,1,n+1}]-Sum[Sum[1/(i+j-1),{i,1,n}],{j,1,n}]]. a(n) = 1/3*Denominator[H(2n+1) + H(2n) - 2H(n)], where H(n) = Sum[1/k, (k, 1, n}] is a Harmonic number, H[n] = A001008/A002805. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

n=2: HilbertMatrix[n,n]

1 1/2

1/2 1/3

so a(1) = 1/3*Denominator[(1 + 1/2 + 1/2 + 1/3) - 1] = 1/3*Denominator[7/3 -1] = 1/3*Denominator[4/3] = 1.

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 ...

Denominator[Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n, 2, 27}]-Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n, 26}]]/3 - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006

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

Sequence in context: A005567 A073391 A002802 this_sequence A122892 A125347 A005465

Adjacent sequences: A101026 A101027 A101028 this_sequence A101030 A101031 A101032

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 17 2004