A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

0, 16, 928, 119344, 3078464, 1132669904, 606887707616, 49610806397296, 48006150564413056, 48265162121607952, 8192066749392160288, 15200753287254377716912, 33677610844789597790454208

Offset: 1

Author

Artur Jasinski, Mar 03 2010

Keywords

• frac
• nonn
• easy

Comments

All numbers in this sequence are divisible by 16. For A173953/16 see A173955.

a(n+2)/A173954(n+2) is, for n >= 0, the partial sum Sum_{k=0..n} 1/(k + 3/4)^2 = 16*Sum_{k=0..n} 1/(4*k + 3)^2. The limit n -> infinity is given in A282824 as Zeta(2, 3/4) = Psi(1, 3/4) = Pi^2 - 8*Catalan, with the trigamma function Psi(1, z) and the Catalan constant A006752.

Examples

			The rationals r(n) = Zeta(2, 3/4) - Zeta(2, n-1/4) begin:  0/1, 16/9, 928/441, 119344/53361, 3078464/1334025, 1132669904/481583025, 606887707616/254757420225, 49610806397296/20635351038225, ... - _Wolfdieter Lang_, Nov 14 2017
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250
• Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
• Eric Weisstein's World of Mathematics, Trigamma Function

Crossrefs

Denominators are in A173954.

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

Programs

Magma

[0] cat [Numerator((&+[16/(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(numer(simplify(r(n))), n=1..13); # Peter Luschny, Nov 14 2017

Mathematica

Table[Numerator[FunctionExpand[Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)

PARI

for(n=1,20, print1(numerator(16*sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

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

Numerator of 128*n*Sum_{k>=1} (4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2). - Vaclav Kotesovec, Nov 14 2017

Numerator of 16*Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. See a comment above. - Wolfdieter Lang, Nov 14 2017

Extensions

Name simplified by Peter Luschny, Nov 14 2017