A382813 - OEIS

%I #29 Apr 25 2025 17:15:32

%S 1,4,18,144,200,400,4900,78400,635040,6350400,64033200,768398400,

%T 9275666400,8657288640,16232416200,519437318400,2779951574400,

%U 16679709446400,60213751101504,3823095308032,1216439416192,26761667156224,1769615240705312,127412297330782464,3062795608913040000

%N Denominator of the n-th partial sum of the squares of the harmonic numbers.

%C All terms for n>1 are even.

%H Robert Israel, <a href="/A382813/b382813.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = denominator((n+1)*H(n)^2-(2*n+1)*H(n)+2*n), where H(n) is the n-th harmonic number.

%F a(n) = denominator((S(n)*H(n)^2+(2*n-2*S(n)+1)*H(n) - 2*n)/(H(n) - 1)), where S(n) = the n-th partial sum of H(n).

%e The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3) = 18.

%p H2:= n-> add(harmonic(k)^2, k = 1..n): seq(denom(H2(n)), n=1..25);

%o (PARI) a(n) = denominator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ _Michel Marcus_, Apr 07 2025

%Y Cf. A001008, A002805, A382812 (numerators).

%Y Cf. A027611, A027612.

%K nonn,frac

%O 1,2

%A _Gary Detlefs_, Apr 05 2025