A276594 -id:A276594 - cp's OEIS frontend

A276592 Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

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

Offset: 1

Martin Renner, Sep 07 2016

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

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.

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

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

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 *)

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

A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

Original entry on oeis.org

8, 96, 960, 161280, 2903040, 638668800, 49816166400, 83691159552000, 2845499424768000, 1946321606541312000, 408727537373675520000, 48662619743783485440000, 124089680346647887872000000, 174221911206693634572288000000, 70734095949917615636348928000000

Offset: 1

Martin Renner, Sep 07 2016

A276592(n)/a(n) * Pi^(2*n) = Sum_{k>=1} 1/(2*k-1)^(2*n) > 1. So Pi^(2*n) > a(n)/A276592(n). - Seiichi Manyama, Sep 03 2018

			From _Seiichi Manyama_, Sep 03 2018: (Start)
n |    Pi^(2*n)   |   a(n)/A276592(n)
--+---------------+------------------------------------
1 |        9.8... |           8
2 |       97.4... |          96
3 |      961.3... |         960
4 |     9488.5... |      161280/17     =     9487.0...
5 |    93648.0... |     2903040/31     =    93646.4...
6 |   924269.1... |   638668800/691    =   924267.4...
7 |  9122171.1... | 49816166400/5461   =  9122169.2... (End)

Seiichi Manyama, Table of n, a(n) for n = 1..225

Cf. A002432, A046988, A276592, A276594, A276595.

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

a[n_]:=Denominator[(1-2^(-2 n)) Zeta[2 n]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Denominator[1/2 SeriesCoefficient[1/(E^x+1),{x,0,2 n-1}]] (* Steven Foster Clark, Mar 10 2023 *)
a[n_]:=Denominator[1/2 Residue[Zeta[s] Gamma[s] (1-2^(1-s)) x^(-s),{s,1-2 n}]] (* Steven Foster Clark, Mar 11 2023 *)

A276592(n)/a(n) + A276594(n)/A276595(n) = A046988(n)/A002432(n).

A276592(n)/a(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

A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

Original entry on oeis.org

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

Offset: 1

Martin Renner, Sep 07 2016

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

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

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

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

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

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

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

cp's OEIS Frontend

A276592 Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

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A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

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A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

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cp's OEIS Frontend

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).

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Formula

A276593 Denominator 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).

Views

Author

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Crossrefs

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Maple

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Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A276592 Numerator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

A276593 Denominator of the rational part of the sum of reciprocals of even powers of odd numbers, i.e., Sum_{k>=1} 1/(2k-1)^(2n).

A276595 Denominator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).