A101029 Denominator of partial sums of a certain series.
1, 10, 70, 420, 4620, 60060, 60060, 408408, 7759752, 38798760, 892371480, 4461857400, 13385572200, 55454513400, 1719089915400, 3438179830800, 24067258815600, 890488576177200, 890488576177200, 36510031623265200, 1569931359800403600, 1569931359800403600, 73786773910618969200
Offset: 1
Examples
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(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 ...
Links
- Eric Weisstein's World of Mathematics, Hilbert Matrix.
- Eric Weisstein's World of Mathematics, Harmonic Number.
Programs
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Mathematica
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 *) -
PARI
a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ Michel Marcus, Feb 28 2022
Formula
a(n) = denominator(s(n)) with s(n) = 3*Sum_{k=1..n} 1/((2*k-1)*k*(2*k+1)). See A101028 for more information.
a(n) = (1/3)*denominator((Sum_{i=1..n+1} Sum_{j=1..n+1} 1/(i+j-1)) - (Sum_{i=1..n} Sum_{j=1..n} 1/(i+j-1))). a(n) = (1/3)*denominator(H(2*n+1) + H(2*n) - 2*H(n)), where H(n) = Sum_{k=1..n} 1/k is a harmonic number, H(n) = A001008/A002805. - Alexander Adamchuk, Apr 11 2006
Extensions
More terms from Michel Marcus, Feb 28 2022
Comments