A173954 a(n) = denominator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

1, 9, 441, 53361, 1334025, 481583025, 254757420225, 20635351038225, 19830572347734225, 19830572347734225, 3351366726767084025, 6196677077792338362225, 13688459664843275442155025

Offset: 1

Author

Artur Jasinski, Mar 03 2010

Keywords

• frac
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..300

Crossrefs

For numerators see A173953.

The Catalan constant is in A006752.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955.

Programs

Magma

[1] cat [Denominator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

Maple

r := n -> Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4):
seq(denom(simplify(r(n))), n=1..13); # Peter Luschny, Nov 14 2017

Mathematica

Table[Denominator[FunctionExpand[-8*Catalan + Pi^2 - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Denominator[Table[8*n*Sum[(-1 + 4*k + 2*n) / ((-1 + 4*k)^2*(-1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Denominator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* G. C. Greubel, Aug 23 2018 *)

PARI

for(n=1,20, print1(denominator(sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

a(n) = denominator of (Pi^2 - 8*Catalan - Zeta(2, (4 n - 1)/4)).

a(n) = denominator of Sum_{k=0..(n-2)} 1/(4*k+3)^2. - G. C. Greubel, Aug 23 2018

Extensions

Name simplified by Peter Luschny, Nov 14 2017