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 A382812 #35 Aug 10 2025 11:38:12 %S A382812 1,13,119,1577,3233,8867,141563,2844129,28119709,335676251,3968696491, %T A382812 55023970333,758025067309,799020611041,1676892996083,59597395635137, %U A382812 351844709221043,2314823924364859,9114392136427625,628176680098075,216039223801697,5117413095318143,363066107054194281,27957386425926920257 %N A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers. %H A382812 Robert Israel, <a href="/A382812/b382812.txt">Table of n, a(n) for n = 1..1000</a> %F A382812 a(n) = numerator((n+1)*H(n)^2-(2*n+1)*H(n) + 2*n), where H(n) is the n-th harmonic number. %F A382812 a(n) = numerator((S(n)*H(n)^2 + (2*n - 2*S(n) + 1)*H(n)-2*n)/(H(n)-1)), where S(n) is the n-th partial sum of H(n). %e A382812 The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3)=119. %p A382812 H2:= n-> add(harmonic(k)^2, k = 1..n): seq(numer(H2(n)), n=1..25); %t A382812 Accumulate[HarmonicNumber[Range[30]]^2]//Numerator (* _Harvey P. Dale_, Aug 10 2025 *) %o A382812 (PARI) a(n) = numerator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ _Michel Marcus_, Apr 07 2025 %Y A382812 Cf. A001008, A002805, A382813 (denominators). %Y A382812 Cf. A027611, A027612 %K A382812 nonn,frac %O A382812 1,2 %A A382812 _Gary Detlefs_, Apr 05 2025