A120076 Numerators of row sums of rational triangle A120072/A120073.

3, 37, 169, 4549, 4769, 241481, 989549, 9072541, 1841321, 225467009, 227698469, 38801207261, 39076419341, 196577627041, 790503882349, 229526961468061, 230480866420061, 83512167402400421, 3351610394325821

Offset: 2

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

The corresponding denominators are given by A120077.

See the W. Lang link under A120072 for more details.

Examples

			The rationals a(m)/A120077(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A120070, A120072, A120073, A120074, A120075, A120077.

Programs

Magma

A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120076(n): n in [2..30]]; // G. C. Greubel, Apr 24 2023

Mathematica

Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)

SageMath

def A120076(n): return numerator(harmonic_number(n,2) - 1/n)
[A120076(n) for n in range(2,31)] # G. C. Greubel, Apr 24 2023

Formula

a(n) = numerator(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 in A103345.

O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).