A276593 -id:A276593

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A276592

Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2*k-1)^(2*n).

+0
6

1, 1, 1, 17, 31, 691, 5461, 929569, 3202291, 221930581, 4722116521, 56963745931, 14717667114151, 2093660879252671, 86125672563201181, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 14129659550745551130667441, 16103843159579478297227731

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Apart from signs, same as A089171 and A279370. - Peter Bala, Feb 07 2019

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..276

Siddharth Dwivedi, Vivek Kumar Singh, and Abhishek Roy, Semiclassical limit of topological Rényi entropy in 3d Chern-Simons theory, arXiv:2007.07033 [hep-th], 2020. See also J. of High Energy Physics (2020) Vol. 2020, Issue 12, Article 132.

FORMULA

a(n)/A276593(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

a(n)/A276593(n) = (-1)^(n+1) * B_{2*n} * (2^(2*n) - 1) / (2 * (2*n)!), where B_n is the Bernoulli number. - Seiichi Manyama, Sep 03 2018

MAPLE

seq(numer(sum(1/(2*k-1)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..22);

MATHEMATICA

a[n_]:=Numerator[Pi^(-2 n) (1-2^(-2 n)) Zeta[2 n]] (* Steven Foster Clark, Mar 10 2023 *)

a[n_]:=Numerator[(-1)^n SeriesCoefficient[1/(E^x+1), {x, 0, 2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)

a[n_]:=Numerator[(-1)^n Residue[Zeta[s] Gamma[s] (1-2^(1-s)), {s, 1-2 n}]] (* Steven Foster Clark, Mar 11 2023 *)

CROSSREFS

Cf. A002432, A046988, A276593, A276594, A276595, A089171, A279370.

KEYWORD

nonn,frac

AUTHOR

Martin Renner, Sep 07 2016

STATUS

approved

A276595

Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).

+0
4

24, 1440, 60480, 2419200, 95800320, 2615348736000, 149448499200, 21341245685760000, 10218188434341888000, 1605715325396582400000, 28202200078783610880000, 3387648273463487338905600000, 372269041039943663616000000, 75786531374911731038945280000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Denominator of Bernoulli(2*n)/(2*(2*n)!). - Robert Israel, Sep 18 2016

LINKS

Robert Israel, Table of n, a(n) for n = 1..223

FORMULA

A276592(n)/A276593(n) + A276594(n)/a(n) = A046988(n)/A002432(n).

Zeta(2n) = (-1)^(n-1)*(A276594(n)/a(n))*((2*Pi)^(2n)), according to Euler. - Terry D. Grant, Jun 19 2018

MAPLE

seq(denom(sum(1/(2*k)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..24);

seq(denom(bernoulli(2*n)/2/(2*n)!), n=1..24); # Robert Israel, Sep 18 2016

MATHEMATICA

Table[Denominator[Zeta[2*n]/(2*Pi)^(2*n)], {n, 1, 30}] (* Terry D. Grant, Jun 19 2018 *)

PROG

(PARI) a(n) = denominator(bernfrac(2*n)/(2*(2*n)!)); \\ Michel Marcus, Jul 05 2018

CROSSREFS

Cf. A002432, A046988, A276592, A276593, A276594.

KEYWORD

nonn,frac

AUTHOR

Martin Renner, Sep 07 2016

STATUS

approved

A276594

Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2*k)^(2*n).

+0
3

1, 1, 1, 1, 1, 691, 1, 3617, 43867, 174611, 77683, 236364091, 657931, 3392780147, 1723168255201, 7709321041217, 151628697551, 26315271553053477373, 154210205991661, 261082718496449122051, 1520097643918070802691, 2530297234481911294093

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Table of n, a(n) for n=1..22.

FORMULA

A276592(n)/A276593(n) + a(n)/A276595(n) = A046988(n)/A002432(n).

MAPLE

seq(numer(sum(1/(2*k)^(2*n), k=1..infinity)/Pi^(2*n)), n=1..24);

CROSSREFS

Cf. A002432, A046988, A276592, A276593, A276595.

KEYWORD

nonn,frac

AUTHOR

Martin Renner, Sep 07 2016

STATUS

approved

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