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Search: id:A117664
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| A117664 |
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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). |
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3
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| 1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 (alex(AT)kolmogorov.com), Apr 23 2006
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Harmonic Number
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FORMULA
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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 (alex(AT)kolmogorov.com), Apr 23 2006
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EXAMPLE
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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 ...
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MATHEMATICA
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Table[Denominator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 30}]
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CROSSREFS
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Cf. A091342, A098118, A111876, A082687, A086881, A005249, A001008, A002805.
Numerator is A117731(n).
Sequence in context: A023372 A025541 A083108 this_sequence A091342 A093454 A048531
Adjacent sequences: A117661 A117662 A117663 this_sequence A117665 A117666 A117667
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KEYWORD
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nonn
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AUTHOR
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Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 11 2006
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