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A117664 Denominator of the sum of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n). 3
1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

n*a(n) = A111876(n-1)

Sum[Sum[1/(i+j-1), {i, 1, n}], {j, 1, n}]] = 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

LINKS

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, Hilbert Matrix.

Eric Weisstein's World of Mathematics, Harmonic Number

FORMULA

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

a(n) = Denominator[Sum[Sum[1/(i+j-1), {i, 1, n}], {j, 1, n}]]. Numerator is A117731(n). - Alexander Adamchuk, Apr 23 2006

EXAMPLE

n=2: HilbertMatrix[n,n]

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 ...

MATHEMATICA

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

CROSSREFS

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

Numerator is A117731(n).

Sequence in context: A233257 A261127 A083108 * A091342 A093454 A048531

Adjacent sequences:  A117661 A117662 A117663 * A117665 A117666 A117667

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Apr 11 2006

STATUS

approved

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Last modified April 18 16:45 EDT 2019. Contains 322209 sequences. (Running on oeis4.)