login
Denominator of the n-th partial sum of the generalized harmonic numbers A007406/A007407.
2

%I #17 Jun 09 2024 23:58:54

%S 1,4,18,144,600,3600,44100,78400,635040,254016,12806640,153679680,

%T 1855133280,8657288640,16232416200,519437318400,8339854723200,

%U 150117385017600,541923759913536,516117866584320

%N Denominator of the n-th partial sum of the generalized harmonic numbers A007406/A007407.

%C Partial sums of A007406/A007407 are 1, 9/4, 65/18, 725/144, 3899/600, 28763/3600, ...

%p A007406_7 := proc(n)

%p local i;

%p add(1/i^2,i=1..n) ;

%p end proc:

%p A370774 := proc(n)

%p add( A007406_7(i),i=1..n) ;

%p denom(%) ;

%p end proc:

%p seq(A370774(n),n=1..20) ;

%t Table[-EulerGamma + HarmonicNumber[1 + n, 2] + n*HarmonicNumber[1 + n, 2] - PolyGamma[0, 2 + n], {n, 1, 20}] // Denominator (* _Vaclav Kotesovec_, May 02 2024 *)

%o (PARI) a(n) = denominator(sum(k=1, n, sum(i=1, k, 1/i^2))); \\ _Michel Marcus_, May 01 2024

%o (Python)

%o from fractions import Fraction

%o def A370774(n): return sum(Fraction(n-i+1,i**2) for i in range(1,n+1)).denominator # _Chai Wah Wu_, May 01 2024

%Y Cf. A120286 (numerators).

%K nonn,frac,easy

%O 1,2

%A _R. J. Mathar_, May 01 2024