A091342 -id:A091342 - cp's OEIS frontend

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

Alexander Adamchuk, Apr 11 2006

nonn

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

			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 ...
  ...

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

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

Numerator is A117731(n).

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

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

A097382 a(h) = d(h,j) = lcm( f(h,j,1) ... f(h,j,h) ), when j=2.

Original entry on oeis.org

1, 4, 35, 20, 3003, 560, 692835, 4620, 185910725, 48048, 136745788725, 1361360, 4281195077775, 22170720, 6541380665835015, 446185740, 1898924328582105825, 5949143200, 90048990529077755175

Offset: 1

Scott C Macfarlan (scottmacfarlan(AT)covance.com), Sep 18 2004

nonn

See A091342, A025558

			a(2) = 4 = lcm(4,1)
a(5) = 3003 = lcm(13,11,3,7,1)
a(6) = 560 = lcm(16,7,4,5,8,1)
a(13) = 4281195077775 = lcm(37,35,11,31,29,9,25,23,7,19,17,5,1)

Given (1) f(h, j, a) = ( [ ((a/gcd(a, h)) / gcd(j+1, (a/gcd(a, h)))) * (h(j+1)) ] - [ ((a/gcd(a, h)) / gcd(j+1, (a/gcd(a, h)))) * (ja) ]) / a then let (2) a(h) = d(h, j) = lcm( f(h, j, 1) ... f(h, j, h) )

A098135 a(h) = d(h,j) = lcm( f(h,j,1) ... f(h,j,h) ), when j=3.

Original entry on oeis.org

1, 5, 9, 455, 2618, 315, 271700, 56751695, 6930, 3708514810, 32152414840, 135135, 19749267715100, 314645828225300, 34918884, 113543477411038675, 473930651619825400, 669278610, 467655703593591713200

Offset: 0

Scott C. Macfarlan (scottmacfarlan(AT)covance.com), Sep 27 2004

nonn

			a(2) = 5 = lcm(5,1)
a(3) = 9 = lcm(9,3,1)
a(5) = 2618 = lcm(17,7,11,2,1)
a(11) = 32152414840 = (41,19,35,8,29,13,23,5,17,7,1)

Cf. A091342, A025558, A097382.

Given f(h, j, a) = ( [ ((a/gcd(a, h)) / gcd(j+1, (a/gcd(a, h)))) * (h(j+1)) ] - [ ((a/gcd(a, h)) / gcd(j+1, (a/gcd(a, h)))) * (ja) ]) / a then let a(h) = d(h, j) = lcm( f(h, j, 1) ... f(h, j, h) )

cp's OEIS Frontend

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.

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A097382 a(h) = d(h,j) = lcm( f(h,j,1) ... f(h,j,h) ), when j=2.

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Formula

A098135 a(h) = d(h,j) = lcm( f(h,j,1) ... f(h,j,h) ), when j=3.

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Author

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