A173947 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

0, 16, 416, 34096, 5794624, 1680121936, 82501802464, 2065646660464, 1739147340740224, 210617970218777104, 288533264855755545376, 485294472126860897387056, 485518650207447822251456

Offset: 0

Author

Artur Jasinski, Mar 03 2010

Keywords

• frac
• nonn
• easy

Comments

For A173947/16 see A173949.

a(n+1)/A173948(n+1) =: r(n) = (Zeta(2, 1/4) - Zeta(2, n + 5/4)), the partial sum Sum_{k=0..n} 1/(k + 1/4)^2, n >= 0. The limit is Zeta(2, 1/4) = A282823 = 16*A222183. - Wolfdieter Lang, Nov 14 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
• Eric Weisstein's World of Mathematics, Trigamma Function

Crossrefs

Cf. A006752, A120268, A173945, A173948 (denominators), A173949.

Programs

Magma

[1] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 22 2018

Maple

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

Mathematica

Table[Numerator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(1 + 4*k + 2*n) / ((1 + 4*k)^2*(1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 1)^2, {k, 0, n - 1} ], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

PARI

for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

Formula

a(n) = numerator of 8*Catalan + Pi^2 - Zeta(2, (4 n + 1)/4), with the Catalan constant given in A006752.

a(n) = numerator(r(n)) with r(n) = Zeta(2, 1/4) - Zeta(2, n + 1/4), with the Hurwitz Zeta function (see the name). With Zeta(2, 1/4) = Psi(1, 1/4) = 8*Catalan + Pi^2 this is the preceding formula, where Psi(1, z) is the Trigamma function. - Wolfdieter Lang, Nov 14 2017

Extensions

Name simplified and offset set to 0 by Peter Luschny, Nov 14 2017