

A173955


a(n) = numerator of (Zeta(2, 3/4)  Zeta(2, n1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.


11



0, 1, 58, 7459, 192404, 70791869, 37930481726, 3100675399831, 3000384410275816, 3016572632600497, 512004171837010018, 950047080453398607307, 2104850677799349861903388, 609822785846772474028096357, 611130542819711220012487366
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OFFSET

1,3


COMMENTS

The denominators are given in A173954.
a(n+2)/A173954(n+2) = (Zeta(2, 3/4)  Zeta(2, n + 7/4))/16 gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(4*n + 3). In the limit n > infinity the series value is Zeta(2,3/4)/16, with the Hurwitz Zeta function, and it is given in A247037.  Wolfdieter Lang, Nov 15 2017


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..250


FORMULA

a(n) = numerator of r(n) with r(n) = (Pi^2  8*Catalan  Zeta(2, n  1/4))/16, with the Hurwitz Zeta function Z(2, z), and the Catalan constant is given in A006752. With Zeta(2, 3/4) = Pi^2  8*Catalan this is the formula given in the name.
Numerator of Sum_{k=0..n2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0.  G. C. Greubel, Aug 23 2018


MAPLE

r := n > (Zeta(0, 2, 3/4)  Zeta(0, 2, n1/4))/16:
seq(numer(simplify(r(n))), n=1..15); # Peter Luschny, Nov 14 2017


MATHEMATICA

Table[Numerator[FunctionExpand[(Pi^2  8*Catalan  Zeta[2, (4*n  1)/4])/16]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[8*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[Sum[1/(4*k + 3)^2, {k, 0, n2}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)


PROG

(PARI) for(n=1, 20, print1(numerator(sum(k=0, n2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(MAGMA) [0] cat [Numerator((&+[1/(4*k+3)^2: k in [0..n2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018


CROSSREFS

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A247037.
Sequence in context: A282438 A308391 A128934 * A243466 A201988 A116103
Adjacent sequences: A173952 A173953 A173954 * A173956 A173957 A173958


KEYWORD

frac,nonn,easy


AUTHOR

Artur Jasinski, Mar 03 2010


EXTENSIONS

Numbers changed according to the old (or new) Mathematica program, and edited by Wolfdieter Lang, Nov 14 2017
Name simplified by Peter Luschny, Nov 14 2017


STATUS

approved



