A382813 Denominator of the n-th partial sum of the squares of the harmonic numbers.

1, 4, 18, 144, 200, 400, 4900, 78400, 635040, 6350400, 64033200, 768398400, 9275666400, 8657288640, 16232416200, 519437318400, 2779951574400, 16679709446400, 60213751101504, 3823095308032, 1216439416192, 26761667156224, 1769615240705312, 127412297330782464, 3062795608913040000

Offset

1,2

Comments

All terms for n>1 are even.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Formula

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

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).

Example

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

Maple

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

Prog

(PARI) a(n) = denominator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

Crossrefs

Cf. A001008, A002805, A382812 (numerators).

Cf. A027611, A027612.

Sequence in context: A214647 A156445 A304997 * A370774 A060841 A059837

Adjacent sequences: A382810 A382811 A382812 * A382814 A382815 A382816

Keyword

nonn,frac

Author

Gary Detlefs, Apr 05 2025

Status

approved