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A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function. 11
0, 16, 928, 119344, 3078464, 1132669904, 606887707616, 49610806397296, 48006150564413056, 48265162121607952, 8192066749392160288, 15200753287254377716912, 33677610844789597790454208 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

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

EXAMPLE

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

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 *)

PROG

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

(MAGMA) [0] cat [Numerator((&+[16/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

CROSSREFS

Denominators are in A173954.

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

Sequence in context: A006089 A260620 A290940 * A211105 A276637 A211081

Adjacent sequences:  A173950 A173951 A173952 * A173954 A173955 A173956

KEYWORD

frac,nonn,easy

AUTHOR

Artur Jasinski, Mar 03 2010

EXTENSIONS

Name simplified by Peter Luschny, Nov 14 2017

STATUS

approved

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Last modified January 18 14:04 EST 2020. Contains 331011 sequences. (Running on oeis4.)