This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A101029 #17 Feb 16 2025 08:32:55
%S A101029 1,10,70,420,4620,60060,60060,408408,7759752,38798760,892371480,
%T A101029 4461857400,13385572200,55454513400,1719089915400,3438179830800,
%U A101029 24067258815600,890488576177200,890488576177200,36510031623265200,1569931359800403600,1569931359800403600,73786773910618969200
%N A101029 Denominator of partial sums of a certain series.
%C A101029 The numerators are given in A101028.
%C A101029 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
%H A101029 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%H A101029 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F A101029 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.
%F A101029 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
%e A101029 n=2: HilbertMatrix[n,n]
%e A101029 1 1/2
%e A101029 1/2 1/3
%e A101029 so a(1) = (1/3)*denominator((1 + 1/2 + 1/2 + 1/3) - 1) = (1/3)*denominator(4/3) = 1.
%e A101029 The n X n Hilbert matrix begins:
%e A101029 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e A101029 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e A101029 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e A101029 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e A101029 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e A101029 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%t A101029 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 *)
%o A101029 (PARI) a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ _Michel Marcus_, Feb 28 2022
%Y A101029 Cf. A098118, A086881, A005249, A001008, A002805.
%Y A101029 Cf. A101028 (numerators).
%K A101029 nonn,frac,easy
%O A101029 1,2
%A A101029 _Wolfdieter Lang_, Dec 17 2004
%E A101029 More terms from _Michel Marcus_, Feb 28 2022