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Denominator of Sum_{k = 0..n} 1/((k+1)*(2*k+1)).
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%I #22 Jul 24 2023 02:38:08

%S 1,6,30,420,1260,13860,180180,72072,1225224,116396280,116396280,

%T 2677114440,13385572200,5736673800,166363540200,10314539492400,

%U 10314539492400,72201776446800,2671465728531600,2671465728531600

%N Denominator of Sum_{k = 0..n} 1/((k+1)*(2*k+1)).

%H G. C. Greubel, <a href="/A111876/b111876.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = denominator of ( digamma(n+3/2) - digamma(n+2) + 2*log(2) ).

%F a(n) = denominator of 2*(n+1)*Integral_{x = 0..1} x^n* log(1+sqrt(x)) dx.

%F a(n-1) = denominator( (1/n)*Sum_{k = 1..n} (n - k)/(n + k) ). - _Peter Bala_, Oct 10 2021

%p seq(denom( add(1/((k+1)*(2*k+1)), k = 0..n) ), n = 0..20); # _Peter Bala_, Oct 10 2021

%t Table[Denominator[HarmonicNumber[2n+2] - HarmonicNumber[n+1]]/2, {n, 0, 30}]

%o (PARI) a(n) = denominator(sum(k=0, n, 1/((k+1)*(2*k+1)))); \\ _Michel Marcus_, Oct 10 2021

%o (Magma) [Denominator(HarmonicNumber(2*n+2) -HarmonicNumber(n+1))/2: n in [0..40]]; // _G. C. Greubel_, Jul 24 2023

%o (SageMath) [denominator(harmonic_number(2*n+2,1) - harmonic_number(n+1,1))/2 for n in range(41)] # _G. C. Greubel_, Jul 24 2023

%Y Cf. A082687 (numerators), A117664.

%K easy,nonn,frac

%O 0,2

%A _Paul Barry_, Aug 19 2005