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A101029
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Denominator of partial sums of a certain series.
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1
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1, 10, 70, 420, 4620, 60060, 60060, 408408, 7759752, 38798760, 892371480, 4461857400, 13385572200, 55454513400, 1719089915400, 3438179830800, 24067258815600, 890488576177200, 890488576177200, 36510031623265200, 1569931359800403600, 1569931359800403600, 73786773910618969200
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OFFSET
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1,2
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COMMENTS
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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, Apr 11 2006
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LINKS
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FORMULA
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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, Apr 11 2006
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EXAMPLE
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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 ...
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MATHEMATICA
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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, Apr 11 2006 *)
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PROG
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(PARI) a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ Michel Marcus, Feb 28 2022
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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