A002432 -id:A002432 - cp's OEIS frontend

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).

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

A100594 Floor of Pi^(2n)/Zeta(2n).

Original entry on oeis.org

6, 90, 945, 9450, 93555, 924041, 9121612, 90030844, 888579011, 8769948429, 86555983552, 854273468992, 8431341566236, 83214006759229, 821289329637860, 8105800788023426, 80001047145799660, 789578687036411293

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 30 2004

nonn

			a(1)=6 because Zeta(2*1)=Pi^2/6 implies Pi^2/Zeta(2)=6 and floor(6)=6.
a(6)=924041 because Zeta(2*6)=691/638512875*Pi^12 implies Pi^12/Zeta(12)=638512875/691 and floor(638512875/691)=924041.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A002432, A046988.

seq(simplify(floor(Pi^(2*k)/Zeta(2*k))),k=1..24);

Table[Floor[Pi^(2*n)/Zeta[2*n]],{n,20}] (* Terry D. Grant, May 28 2017 *)

{a(n)=if(n<1, 0, floor(-2*(2*n)!/(-4)^n/bernfrac(2*n)))} /* Michael Somos, Feb 18 2007 */

A104007 Denominators of coefficients in expansion of x^2(1-exp(-2x))^(-2).

Original entry on oeis.org

4, 2, 12, 6, 60, 90, 378, 945, 2700, 9450, 20790, 93555, 116093250, 638512875, 1403325, 18243225, 43418875500, 325641566250, 4585799468250, 38979295480125, 161192575293750, 1531329465290625, 640374140030625, 13447856940643125, 17558223649022306250

Offset: 0

N. J. A. Sloane, Apr 17 2005

It appears that a(2n+2) = A002432(n). As A098087(n)/A104007(n) = x*(csch(x)^4)/(4*(coth(x)-1)^2), then a(2n+2) would represent the sequence of denominators for just the even powers of the full series representation at x=0. A002432 could be conjectured to be the non-hyperbolic, or circle trigonometric, function equivalent where the full series of denominators could be found by the formula x*((csc(x)^2)/4) - cot(x)/2) + 1 for a(n) > 4.
Hyperbolic Trigonometric Functions:
Entire Series: x*(csch(x)^4) / (4*(coth(x)-1)^2).
Even Powers: (1/2)*(1-x*coth(x)).
Odd Powers: (1/4)*(2x + (csch(x)^2) + 2).
Circular Trigonometric Functions:
Entire Series: x*((csc(x)^2)/4) - cot(x)/2) + 1.
Even Powers: (1/2)*(1-x*cot(x)).
Odd Powers: (1/4)*(2x + (csc(x)^2) + 2).
In turn, one may be able to derive some constant for x that can represent the zeta functions of odd positive integers. For zeta functions of even positive integers, that constant is Pi. - Terry D. Grant, Sep 24 2016
One can use the connection of the expansion of x^2*(1-exp(-2*x))^(-2) to Bernoulli numbers to prove that a(2n+2) = A002432(n), a(2n) = denominator(zeta(2n-2)) and a(2n-1) = denominator(1/2 (2n-3) zeta(2n-2)), and more generally that the expansion of x^2*(1-exp(-2*x))^(-2) is related to zeta(2n). The connection to Bernoulli numbers comes from the fact that x^2*(1-exp(-2*x))^(-2) is related to the trigonometric functions cot and csc, and they both have the series coefficients related to Bernoulli numbers, which are only related to zeta(2n), zeta functions of even positive integers, and not zeta(2n-1), zeta functions of odd positive integers. Because both a(2n) and a(2n-1) are related to zeta functions of even positive integers, the odd or even terms of this sequence are only related to zeta functions of odd positive integers if zeta(2n) is itself related to zeta(2n-1). - Andrey Mitin, Aug 16 2020

See A098087 for further information.

Cf. A002432.

Denominator[ CoefficientList[ Series[x^2*(1 - E^(-2x))^(-2), {x, 0, 33}], x]] (* Robert G. Wilson v, Apr 20 2005 *)
Denominator[
 Function[{n},
   Piecewise[{{1/2 (-1 + n) Zeta[n], Mod[n, 2] == 0}, {Zeta[-1 + n],
Mod[n, 2] == 1}}]] /@ Range[0, 20]] (* Andrey Mitin, Aug 16 2020 *)

A276594 Numerator 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

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

Offset: 1

Martin Renner, Sep 07 2016

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

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

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

A295231 Numerators of (-1)^(n+1) * (2n)! (2^(2n)+1)/(B_{2n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

Original entry on oeis.org

-4, 15, 765, 61425, 1214325, 95893875, 2615987248875, 298915241625, 10670785663663125, 10218227413637368125, 1605716856726047690625, 56404413605424162403125, 3387648475383059302662121875, 744538093174369303262578125

Offset: 0

Seiichi Manyama, Nov 18 2017

Pi^(2*n) > a(n)/A295232(n) for n > 0.

			Zeta(2) = Pi^2/6   > 1 + 1/2^2, so Pi^2 >    15/2.
Zeta(4) = Pi^4/90  > 1 + 1/2^4, so Pi^4 >   765/8.
Zeta(6) = Pi^6/945 > 1 + 1/2^6, so Pi^6 > 61425/64.

Seiichi Manyama, Table of n, a(n) for n = 0..223

Cf. A002432/A046988, A295232 (denominators).

{a(n) = numerator((-1)^(n+1)*(2*n)!*(2^(2*n)+1)/(bernfrac(2*n)*2^(4*n-1)))}

A295232 Denominator of (-1)^(n+1) * (2n)! (2^(2n)+1)/(B_{2n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

Original entry on oeis.org

1, 2, 8, 64, 128, 1024, 2830336, 32768, 118521856, 11499470848, 183092903936, 651652235264, 3965531409350656, 88306004000768, 1821484971735384064, 7400951301593676906496, 16555640873195841519616, 2604961188466481168384

Offset: 0

Seiichi Manyama, Nov 18 2017

Pi^(2*n) > A295231(n)/a(n) for n > 0.

			Zeta(2) = Pi^2/6   > 1 + 1/2^2, so Pi^2 >    15/2.
Zeta(4) = Pi^4/90  > 1 + 1/2^4, so Pi^4 >   765/8.
Zeta(6) = Pi^6/945 > 1 + 1/2^6, so Pi^6 > 61425/64.

Seiichi Manyama, Table of n, a(n) for n = 0..275

Cf. A002432/A046988, A295231 (numerators).

{a(n) = denominator((-1)^(n+1)*(2*n)!*(2^(2*n)+1)/(bernfrac(2*n)*2^(4*n-1)))}

A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

1, 75, 1, 16875, 24, 1, 221484375, 34560, 18, 1, 116279296875, 116121600, 58320, 39, 1, 12950606689453125, 780337152000, 440899200, 296595, 51, 1, 4861333986053466796875, 8899589151129600, 6666395904000, 68420017575, 663255, 63, 1, 677114376628875732421875

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
           1,            1,             1,              1,                 1
          75,           24,            18,             39,                51
       16875,        34560,         58320,         296595,            663255
   221484375,    116121600,     440899200,    68420017575,       20126472975
116279296875, 780337152000, 6666395904000, 93393323989875, 10382542981248375

Index entries for zeta function.

Cf. A370412 (numerators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 30, if(isfundamental(k), v=concat(v, denominator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, denominator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.denominator() # Robin Visser, Mar 19 2024

T(n, k) = denominator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = denominator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = denominator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 1, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

Original entry on oeis.org

0, 2, 0, 4, 1, 0, 536, 11, 1, 0, 2888, 361, 23, 2, 0, 3302008, 24611, 1681, 116, 4, 0, 12724582576, 2873041, 257543, 267704, 328, 4, 0, 18194938976, 27233033477, 67637281, 3741352, 92656, 88, 1, 0, 875222833138832, 11779156811, 18752521534133, 1156377368, 479214352, 287536, 29, 2, 0

Offset: 0

Thomas Scheuerle, Feb 22 2024

			The array begins:
          0,           0,              0,               0,                 0
          2,           1,              1,               2,                 4
          4,          11,             23,             116,               328
        536,         361,           1681,          267704,             92656
       2888,       24611,         257543,         3741352,         479214352
    3302008,     2873041,       67637281,      1156377368,       14816172016
12724582576, 27233033477, 18752521534133, 753075777246704, 16476431095568992

Index entries for zeta function.

Cf. A370411 (denominators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

\p 700
row(n) = {v=[]; for(k=2, 50, if(isfundamental(k), v=concat(v, numerator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v}
z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n))))
row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, numerator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here

# Only suitable for small n and k
def T(n, k):
    discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)]
    D = sorted(list(set(discs)))[k+1]
    zetaK = QuadraticField(D).zeta_function(1000)
    val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900)
    return val.numerator() # Robin Visser, Mar 19 2024

T(n, k) = numerator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = numerator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = numerator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) -  zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 0, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).

A340471 Denominators of an approximation to zeta(n)/Pi^n.

Original entry on oeis.org

2, 6, 28, 90, 1488, 945, 182880, 9450, 8241408, 93555, 14856307200, 638512875, 1569400842240, 18243225, 5713142135500800, 325641566250, 1096948397364019200, 38979295480125, 6713362606110031872000, 1531329465290625, 408173030347971900211200, 13447856940643125

Offset: 1

Melchor Viso Martinez, Jan 08 2021

			1/2, 1/6, 1/28, 1/90, 5/1488, 1/945, 61/182880, 1/9450, 277/8241408, 1/93555, 50521/14856307200, 691/638512875, ...
Values are approximate for odd indices, exact for even indices:
  zeta(1) ~       1/2             zeta(2) = Pi^2/6
  zeta(3) ~    Pi^3/28            zeta(4) = Pi^4/90
  zeta(5) ~  5*Pi^5/1488          zeta(6) = Pi^6/945
  zeta(7) ~ 61*Pi^7/182880,       zeta(8) = Pi^8/9450
  ...

Melchor Viso Martinez, An expression for integer zeta approximation

Cf. A000831, A002432, A331839, A340472 (numerators).

a[k_] := Denominator[(1/(4 (1 - 2^-k) k!)
      D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
       k}]) /. {\[Lambda] -> 0}]

a(n) = {my(t=tan(x/4 + O(x*x^n))); denominator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021

a(n) = denominator of lim_{x->0} of the n-th derivative of x*tan((Pi+x)/4)/((4-2^(2-n))*n!) with respect to x.
a(2*n) = A002432(n).
From Andrew Howroyd, Jan 10 2021: (Start)
a(n) = denominator of (1/(4-2^(2-n)))*[x^n] x*(1 + tan(x/4))/(1 - tan(x/4)).
a(n) = denominator( A000831(n-1)/((n-1)!*2^n*(2^n-1)) ). (End)

A351806 Denominator of zeta({6}_n)/Pi^(6*n).

Original entry on oeis.org

1, 945, 212837625, 64965492466875, 432684797065192546875, 1347828286825972065254765625, 197885500589205605585596463448046875, 18132629348577543860598956218936672646484375, 3673787208165374996876652878250276546299488037109375

Offset: 0

Roudy El Haddad, Feb 19 2022

({6}_n) is standard notation for multiple zeta values. It represents (6, ..., 6) where the multiplicity of 6 is n.

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. A351864 (numerators).
Cf. A002432 (denominators of zeta(2*n)/Pi^(2*n)).
Cf. A013664 (zeta(6)).
Cf. A103345.

a[n_] := Denominator[6*2^(6*n)/(6*n + 3)!]; Array[a, 9, 0] (* Amiram Eldar, Feb 19 2022 *)

a(n) = denominator(6*2^(6*n)/(6*n + 3)!); \\ Michel Marcus, Feb 22 2022

a(n) = denominator(6*2^(6*n)/(6*n + 3)!).

cp's OEIS Frontend

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).

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A100594 Floor of Pi^(2*n)/Zeta(2*n).

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A104007 Denominators of coefficients in expansion of x^2*(1-exp(-2*x))^(-2).

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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).

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A295231 Numerators of (-1)^(n+1) * (2*n)! * (2^(2*n)+1)/(B_{2*n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

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PARI

A295232 Denominator of (-1)^(n+1) * (2*n)! * (2^(2*n)+1)/(B_{2*n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

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PARI

A370411 Square array T(n, k) = denominator( zeta_r(2*n) * sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

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PARI

Sage

Formula

A370412 Square array T(n, k) = numerator( zeta_r(2*n) * sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

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PARI

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).

A100594 Floor of Pi^(2n)/Zeta(2n).

A104007 Denominators of coefficients in expansion of x^2(1-exp(-2x))^(-2).

A276594 Numerator of the rational part of the sum of reciprocals of even powers of even numbers, i.e., Sum_{k>=1} 1/(2k)^(2n).

A295231 Numerators of (-1)^(n+1) * (2n)! (2^(2n)+1)/(B_{2n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

A295232 Denominator of (-1)^(n+1) * (2n)! (2^(2n)+1)/(B_{2n} * 2^(4*n-1)), where B_{n} is the Bernoulli number.

A370411 Square array T(n, k) = denominator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).

A370412 Square array T(n, k) = numerator( zeta_r(2n) sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).