A120076 - OEIS

%I #13 Apr 25 2023 06:17:48

%S 3,37,169,4549,4769,241481,989549,9072541,1841321,225467009,227698469,

%T 38801207261,39076419341,196577627041,790503882349,229526961468061,

%U 230480866420061,83512167402400421,3351610394325821

%N Numerators of row sums of rational triangle A120072/A120073.

%C The corresponding denominators are given by A120077.

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

%H G. C. Greubel, <a href="/A120076/b120076.txt">Table of n, a(n) for n = 2..1000</a>

%F a(n) = numerator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.

%F 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.

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

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

%t Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* _G. C. Greubel_, Apr 24 2023 *)

%o (Magma)

%o A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;

%o [A120076(n): n in [2..30]]; // _G. C. Greubel_, Apr 24 2023

%o (SageMath)

%o def A120076(n): return numerator(harmonic_number(n,2) - 1/n)

%o [A120076(n) for n in range(2,31)] # _G. C. Greubel_, Apr 24 2023

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

%K nonn,easy,frac

%O 2,1

%A _Wolfdieter Lang_, Jul 20 2006