A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600

Offset: 1

Author

Alexander Adamchuk, Apr 11 2006

Keywords

• nonn

Comments

Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) = A117731(n) / A117664(n) = 2n * H'(2n) = 2n * A058313(2n) / A058312(2n), where H'(2n) is 2n-th alternating sign Harmonic Number. H'(2n) = H(2n) - H(n), where H(n) is n-th Harmonic Number. - Alexander Adamchuk, Apr 23 2006

Examples

			For n=2, the 2 X 2 Hilbert matrix is [1, 1/2; 1/2, 1/3], so a(2) = denominator(1 + 1/2 + 1/2 + 1/3) = denominator(7/3) = 3.
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 ...
  ...
		

Links

• Eric Weisstein's World of Mathematics, Hilbert Matrix.
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A091342, A098118, A111876, A082687, A086881, A005249, A001008, A002805.

Numerator is A117731(n).

Programs

Mathematica

Table[Denominator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 30}]

Formula

a(n) = A111876(n-1)/n.

a(n) = denominator( Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) ). Numerator is A117731(n). - Alexander Adamchuk, Apr 23 2006

a(n) = denominator( Sum_{k=1..n} (2*k)/(n+k) ). - Peter Bala, Oct 10 2021