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

Author

Gary Detlefs, Apr 05 2025

Keywords

• nonn
• frac

Comments

All terms for n>1 are even.

Examples

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

Links

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

Crossrefs

Cf. A001008, A002805, A382812 (numerators).

Cf. A027611, A027612.

Programs

Maple

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

PARI

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

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).