A002432 -id:A002432 - cp's OEIS frontend

A013670 Decimal expansion of zeta(12).

1, 0, 0, 0, 2, 4, 6, 0, 8, 6, 5, 5, 3, 3, 0, 8, 0, 4, 8, 2, 9, 8, 6, 3, 7, 9, 9, 8, 0, 4, 7, 7, 3, 9, 6, 7, 0, 9, 6, 0, 4, 1, 6, 0, 8, 8, 4, 5, 8, 0, 0, 3, 4, 0, 4, 5, 3, 3, 0, 4, 0, 9, 5, 2, 1, 3, 3, 2, 5, 2, 0, 1, 9, 6, 8, 1, 9, 4, 0, 9, 1, 3, 0, 4, 9, 0, 4, 2, 8, 0, 8, 5, 5, 1, 9, 0, 0, 6, 9

Offset: 1

N. J. A. Sloane

			1.0002460865533080482986379980477396709604160884580034045330409521332520...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A013662, A013664, A013666, A013668.

RealDigits[Zeta[12],10,120][[1]] (* Harvey P. Dale, Apr 30 2013 *)

PARI
```
zeta(12) \\ Michel Marcus, Feb 20 2015
```

zeta(12) = 2/3*2^12/(2^12 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^13 ), where p(n) = 7*n^12 + 182*n^10 + 1001*n^8 + 1716*n^6 + 1001*n^4 + 182*n^2 + 7 is a row polynomial of A091043. - Peter Bala, Dec 05 2013
zeta(12) = Sum_{n >= 1} (A010052(n)/n^6) = Sum {n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^6 ). - Mikael Aaltonen, Feb 20 2015
zeta(12) = 691/638512875*Pi^12 (see A002432). - Rick L. Shepherd, May 30 2016
zeta(12) = Product_{k>=1} 1/(1 - 1/prime(k)^12). - Vaclav Kotesovec, May 02 2020

A073747 Decimal expansion of coth(1).

Original entry on oeis.org

1, 3, 1, 3, 0, 3, 5, 2, 8, 5, 4, 9, 9, 3, 3, 1, 3, 0, 3, 6, 3, 6, 1, 6, 1, 2, 4, 6, 9, 3, 0, 8, 4, 7, 8, 3, 2, 9, 1, 2, 0, 1, 3, 9, 4, 1, 2, 4, 0, 4, 5, 2, 6, 5, 5, 5, 4, 3, 1, 5, 2, 9, 6, 7, 5, 6, 7, 0, 8, 4, 2, 7, 0, 4, 6, 1, 8, 7, 4, 3, 8, 2, 6, 7, 4, 6, 7, 9, 2, 4, 1, 4, 8, 0, 8, 5, 6, 3, 0, 2, 9, 4, 6, 7, 9

Offset: 1

Rick L. Shepherd, Aug 07 2002

coth(x) = (e^x + e^(-x))/(e^x - e^(-x)).
Because the continued fraction for coth(1) is all positive odd numbers in sequence, the second Mathematica program below also generates the sequence. - Harvey P. Dale, Oct 15 2011
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.31303528549933130363616124693...

Samuel M. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Hideyuki Ohtsuka, Problem 11853, The American Mathematical Monthly, Vol. 122, No. 7 (2015), p. 700; A Hyperbolic Sine Series, Solutions to Problem 11853 by Tewodros Amdeberhan and Rituraj Nandan, ibid., Vol. 124, No. 5 (2017), p. 469.
Allen Stenger, Experimental math for Math Monthly problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131; alternative link.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A005408 (continued fraction: odd numbers), A073821 (continued fraction exp. is even numbers), A073744 (tanh(1)=1/A073747), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)), A349004.

RealDigits[Coth[1],10,120][[1]] (* or *) RealDigits[ FromContinuedFraction[ Range[1,1001,2]],10,120][[1]] (* Harvey P. Dale, Oct 15 2011 *) (* see Comments, above, for the second program *)

PARI
```
1/tanh(1)
```

Equals 1 + Sum_{n>=1} (2^(2*n)*B(2*n))/(2*n)! = 1 + Sum_{n>=1}  (-1)^(n+1)*2*(A046988(n+1) / A002432(n+1)). - Terry D. Grant, May 30 2017
Equals 1 + BesselI(3/2, 1)/BesselI(1/2, 1). - Terry D. Grant, Jun 18 2018
Equals 1 + Sum_{k>=1} csch(2^k) (Ohtsuka, 2015; Stenger, 2017). - Amiram Eldar, Oct 04 2021

A046988 Numerators of zeta(2n)/Pi^(2n).

Original entry on oeis.org

-1, 1, 1, 1, 1, 1, 691, 2, 3617, 43867, 174611, 155366, 236364091, 1315862, 6785560294, 6892673020804, 7709321041217, 151628697551, 26315271553053477373, 308420411983322, 261082718496449122051, 3040195287836141605382, 5060594468963822588186

Offset: 0

N. J. A. Sloane

Equivalently, numerator of (-1)^(n+1)*2^(2*n-1)*Bernoulli(2*n)/(2*n)!. - Lekraj Beedassy, Jun 26 2003

An old name erroneously included "Numerators of Taylor series expansion of log(x/sin(x))"; that is now submitted as a distinct sequence A283301. - Vladimir Reshetnikov, Mar 04 2017

			Numerator(zeta(0)/Pi^0) = -1. - _Artur Jasinski_, Mar 11 2010

L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 222, series for log(H(x)/x).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.

J.P. Martin-Flatin, Table of n, a(n) for n = 0..250
Masato Kobayashi and Shunji Sasaki, Values of zeta-one functions at positive even integers, arXiv:2202.11835 [math.NT], 2022. See p. 4.
Ellise Parnoff and A. Raghuram, Ramanujan's congruence primes, arXiv:2403.03345 [math.NT], 2024.
I. Song, A recursive formula for even order harmonic series, J. Computational and Appl. Math., 21 (1988), 251-256.
Wolfram Research, Some values of zeta(n)
Wolfram Research, A Formula for Zeta(2n)

Cf. A002432 (denominators), A283301, A266214.

seq(numer(Zeta(2*n)/Pi^(2*n)),n=1..24); # Martin Renner, Sep 07 2016

Table[Numerator[Zeta[2 n]/Pi^(2 n)], {n, 0, 30}] (* Artur Jasinski, Mar 11 2010 *)

A013672 Decimal expansion of zeta(14).

Original entry on oeis.org

1, 0, 0, 0, 0, 6, 1, 2, 4, 8, 1, 3, 5, 0, 5, 8, 7, 0, 4, 8, 2, 9, 2, 5, 8, 5, 4, 5, 1, 0, 5, 1, 3, 5, 3, 3, 3, 7, 4, 7, 4, 8, 1, 6, 9, 6, 1, 6, 9, 1, 5, 4, 5, 4, 9, 4, 8, 2, 7, 5, 5, 2, 0, 2, 2, 5, 2, 8, 6, 2, 9, 4, 1, 0, 2, 3, 1, 7, 7, 4, 2, 0, 8, 7, 6, 6, 5, 9, 7, 8, 2, 9, 7, 1, 9, 9, 8, 4, 6

Offset: 1

N. J. A. Sloane

			1.0000612481350587048292585451051353337474816961691545494827552022528629...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

RealDigits[Zeta[14],10,120][[1]] (* Harvey P. Dale, Dec 19 2014 *)

PARI
```
zeta(14) \\ Michel Marcus, Feb 20 2015
```

zeta(14) = Sum_{n >= 1} (A010052(n)/n^7) = Sum {n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^7 ). - Mikael Aaltonen, Feb 20 2015
zeta(14) = 2/18243225*Pi^14 (see A002432). - Rick L. Shepherd, May 30 2016
zeta(14) = Product_{k>=1} 1/(1 - 1/prime(k)^14). - Vaclav Kotesovec, May 02 2020

A013676 Decimal expansion of zeta(18).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 3, 8, 1, 7, 2, 9, 3, 2, 6, 4, 9, 9, 9, 8, 3, 9, 8, 5, 6, 4, 6, 1, 6, 4, 4, 6, 2, 1, 9, 3, 9, 7, 3, 0, 4, 5, 4, 6, 9, 7, 2, 1, 8, 9, 5, 3, 3, 3, 1, 1, 4, 3, 1, 7, 4, 4, 2, 9, 9, 8, 7, 6, 3, 0, 0, 3, 9, 5, 4, 2, 6, 5, 0, 0, 4, 5, 6, 3, 8, 0, 0, 1, 9, 6, 8, 6, 6, 8, 9, 8, 9, 6, 4

Offset: 1

N. J. A. Sloane

			1.0000038172932649998398564616446219397...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

RealDigits[Zeta[18], 10, 100][[1]] (* Alonso del Arte, Feb 07 2016 *)

PARI
```
zeta(18) \\ Michel Marcus, Feb 12 2016
```

zeta(18) = Sum_{n >= 1} (A010052(n)/n^9) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^9 ). - Mikael Aaltonen, Mar 06 2015
zeta(18) = 43867*Pi^18/38979295480125 = A046988(9)*Pi^18/A002432(9). - Alonso del Arte, Feb 12 2016
zeta(18) = Product_{k>=1} 1/(1 - 1/prime(k)^18). - Vaclav Kotesovec, May 02 2020

A117972 Numerator of zeta'(-2n), n >= 0.

Original entry on oeis.org

1, -1, 3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125

Offset: 0

Eric W. Weisstein, Apr 06 2006

In A160464 the coefficients of the ES1 matrix are defined. This matrix led to the discovery that the successive differences of the ES1[1-2*m,n] coefficients for m = 1, 2, 3, ..., are equal to the values of zeta'(-2n), see also A094665 and A160468. - Johannes W. Meijer, May 24 2009
A048896(n), n >= 1: Numerators of Maclaurin series for 1 - ((sin x)/x)^2,
  a(n), n >= 2: Denominators of Maclaurin series for 1 - ((sin x)/x)^2, the correlation function in Montgomery's pair correlation conjecture. - Daniel Forgues, Oct 16 2011
From Andrey Zabolotskiy, Sep 23 2021: (Start)
zeta'(-2n), which is mentioned in the Name, is irrational. For n > 0, a(n) is the numerator of the rational fraction g(n) = Pi^(2n)*zeta'(-2n)/zeta(2n+1). The denominator is 4*A048896(n-1). g(n) = f(n) for n > 0, where f(n) is given in the Formula section. Also, f(n) = Bernoulli(2n)/z(n)/4 (see Formula section) for all n.
For n = 0, zeta'(0) = -log(2Pi)/2, g(0) can be set to 0 because of the infinite denominator. However, a(0) is set to 1 because it is the numerator of f(0).
It seems that -4*f(n)*alpha_n = A000182(n), where alpha_n = A191657(n, p(n)) / A191658(n, p(n)) [where p(n) = A000041(n)] is the n-th "elementary coefficient" from the paper by Izaurieta et al. (End)

			-1/4, 3/4, -45/8, 315/4, -14175/8, 467775/8, -42567525/16, ...
-zeta(3)/(4*Pi^2), (3*zeta(5))/(4*Pi^4), (-45*zeta(7))/(8*Pi^6), (315*zeta(9))/(4*Pi^8), (-14175*zeta(11))/(8*Pi^10), ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Izaurieta, Ricardo Ramírez and Eduardo Rodríguez, Dirac Matrices for Chern-Simons Gravity, arXiv:1106.1648 [math-ph], 2011-2012.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A000367, A002445, A049606, A046988, A002432, A117973, A048896.
From Johannes W. Meijer, May 24 2009: (Start)
Cf. A160464, A094665 and A160468.
Absolute values equal row sums of A160468. (End)
Cf. A191657, A191658, A000182, A000041.

# Without rational arithmetic
a := n -> (-1)^n*(2*n)!*2^(add(i,i=convert(n,base,2))-2*n);
# Peter Luschny, May 02 2009

Table[Numerator[(2 n)!/2^(2 n + 1) (-1)^n], {n, 0, 30}]

L:taylor(1/x*sin(sqrt(x))^2,x,0,15); makelist(denom(coeff(L,x,n))*(-1)^(n+1),n,0,15); /* Vladimir Kruchinin, May 30 2011 */

a(n) = numerator(f(n)) where f(n) = (2*n)!/2^(2*n + 1)(-1)^n, from the Mathematica code.
From Terry D. Grant, May 28 2017: (Start)
|a(n)| = A049606(2n).
a(n) = -numerator(Bernoulli(2n)/z(n)) where Bernoulli(2n) = A000367(n) / A002445(n) and z(n) = A046988(n) / A002432(n) for n > 0. (End) [Corrected by Andrey Zabolotskiy, Sep 23 2021]

First term added, offset changed and edited by Johannes W. Meijer, May 15 2009

A052277 a(n) = (4n+2)!/2^(2n+1).

Original entry on oeis.org

1, 90, 113400, 681080400, 12504636144000, 548828480360160000, 49229914688306352000000, 8094874872198213459360000000, 2252447502438386084347676160000000, 997586474354936812896742294502400000000, 669959124447288464805194190141921792000000000

Offset: 0

N. J. A. Sloane, Feb 05 2000

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, arXiv:hep-th/9611004, 1996.
Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

Cf. A002432 (denominators of zeta(2*n)/Pi^(2*n)).

Cf. A068447, A067912, A013662 (zeta(4)).

Table[(4n+2)!/2^(2n+1), {n, 0, 10}] (* Amiram Eldar, Feb 25 2022 *)

a(n) = (4*n+2)!/2^(2*n+1); \\ Michel Marcus, Feb 20 2022

sin(x)*sinh(x) = Sum_{n>=0} (-1)^n*x^(4n+2)/a(n). - Benoit Cloitre, Feb 02 2002
a(n) = Pi^(4n)/Zeta({4}n) where ({4}_n) is the standard multiple zeta values notation for (4, ..., 4) where the multiplicity of 4 is n. - _Roudy El Haddad, Feb 19 2022
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (cosh(sqrt(2)) - cos(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = sin(1)*sinh(1). (End)

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).

Original entry on oeis.org

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

A308637 Decimal expansion of Pi^3/Zeta(3).

Original entry on oeis.org

2, 5, 7, 9, 4, 3, 5, 0, 1, 6, 6, 6, 1, 8, 6, 8, 4, 0, 1, 8, 5, 5, 8, 6, 3, 6, 5, 7, 9, 3, 9, 6, 5, 1, 3, 2, 9, 0, 0, 5, 0, 9, 5, 2, 3, 2, 7, 1, 3, 1, 2, 2, 6, 0, 7, 0, 6, 1, 4, 0, 2, 1, 3, 4, 0, 6, 4, 9, 4, 3, 4, 9, 1, 3, 4, 9, 2, 5, 0, 6, 1, 4, 1, 2, 2, 5, 1

Offset: 2

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

-----+---------------------------------
   n |  Zeta(n)
-----+---------------------------------
   2 |     Pi^2  / 6         = A013661.
   3 |     Pi^3  / 25.79...  = A002117.
   4 |     Pi^4  / 90        = A013662.
   5 |     Pi^5  / A309926   = A013663.
   6 |     Pi^6  / 945       = A013664.
   7 |     Pi^7  / A309927   = A013665.
   8 |     Pi^8  / 9450      = A013666.
   9 |     Pi^9  / A309928   = A013667.
  10 |     Pi^10 / 93555     = A013668.
  11 |     Pi^11 / A309929   = A013669.
  12 | 691*Pi^12 / 638512875 = A013670.
...
Cf. A002432, A091925, A276120 (Zeta(3)/Pi^3).

RealDigits[Pi^3/Zeta[3], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^3/zeta(3)
```

Pi^3/Zeta(3) = A091925/A002117.

More terms from Amiram Eldar, Aug 24 2019

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

cp's OEIS Frontend

A013670 Decimal expansion of zeta(12).

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A073747 Decimal expansion of coth(1).

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A046988 Numerators of zeta(2*n)/Pi^(2*n).

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A013672 Decimal expansion of zeta(14).

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A013676 Decimal expansion of zeta(18).

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A117972 Numerator of zeta'(-2n), n >= 0.

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A052277 a(n) = (4n+2)!/2^(2n+1).

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A046988 Numerators of zeta(2n)/Pi^(2n).

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).