OFFSET
1,2
COMMENTS
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
LINKS
Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number.
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
EXAMPLE
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 ...
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 *)
PROG
(PARI) a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ Michel Marcus, Feb 28 2022
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Dec 17 2004
EXTENSIONS
More terms from Michel Marcus, Feb 28 2022
STATUS
approved