OFFSET
2,1
COMMENTS
The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
See the W. Lang link under A120072 for more details.
The corresponding numerators are given by A120076.
The n for which a(n) differs from A007407(n) are given by A309829. - Jeppe Stig Nielsen, Aug 18 2019
LINKS
Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150
FORMULA
a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).
EXAMPLE
The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).
MATHEMATICA
Table[Denominator[HarmonicNumber[n, 2] -1/n], {n, 2, 40}] (* G. C. Greubel, Apr 25 2023 *)
PROG
(PARI) a(n) = denominator(sum(j=1, n-1, 1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019
(PARI) a(n) = denominator(sum(j=1, n, 1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019
(Magma)
A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120077(n): n in [2..30]]; // G. C. Greubel, Apr 25 2023
(SageMath)
def A120077(n): return denominator(harmonic_number(n, 2) - 1/n)
[A120077(n) for n in range(2, 31)] # G. C. Greubel, Apr 25 2023
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jul 20 2006
EXTENSIONS
a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019
STATUS
approved