A120077 Denominators of row sums of rational triangle A120072/A120073.

4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 2167695039654144, 1548353599752960, 221193371393280, 117011293467045120

Offset: 2

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

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

Examples

			The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).
		

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150

Crossrefs

Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.

Programs

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

Mathematica

Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)

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

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

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).

Extensions

a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019