A046988 -id:A046988 - cp's OEIS frontend

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).

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).

A231327 Denominator of rational component of zeta(4n)/zeta(2n).

Original entry on oeis.org

1, 15, 105, 675675, 34459425, 16368226875, 218517792968475, 30951416768146875, 694097901592400930625, 23383376494609715287281703125, 2289686345687357378035370971875, 219012470258383844016431785453125, 4791965046290912124048163518904807546875

Offset: 0

Leo Depuydt, Nov 07 2013

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).
For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.
The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). - Leo Depuydt, Nov 22 2013
Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.

Cf. A231273 (the corresponding numerator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, and A156036 (same number sequence as found in numerator, though in various transformations (alternation of sign, intervening numbers, and so on)).
Cf. A027641 and A027642.

seq(denom(bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Denominator[Table[Zeta[4 n]/Zeta[2 n], {n, 0, 15}]] (* T. D. Noe, Nov 15 2013 *)

A283301 Numerators of coefficients at even powers in Taylor series expansion of log(x/sin(x)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 691, 2, 3617, 43867, 174611, 155366, 236364091, 1315862, 3392780147, 6892673020804, 7709321041217, 151628697551, 26315271553053477373, 308420411983322, 261082718496449122051, 3040195287836141605382, 2530297234481911294093

Offset: 0

Vladimir Reshetnikov, Mar 04 2017

This sequence shares many terms with A046988 (and appears to have been erroneously confused with it), but actually differs from it at indexes 0, 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, ...

			log(x/sin(x)) = (1/6)*x^2 + (1/180)*x^4 + (1/2835)*x^6 + (1/37800)*x^8 + (1/467775)*x^10 + (691/3831077250)*x^12 + ...

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.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:4 at page 301.

Cf. A046989 (denominators), A046988.

a[0] = 0; a[n_] := Numerator[((-1)^(n + 1) 2^(2 n - 1) BernoulliB[2 n])/(n (2 n)!)]; Table[a[n], {n, 0, 20}] (* or *)
Numerator@Table[SeriesCoefficient[Log[x/Sin[x]], {x, 0, 2n}], {n, 0, 20}]

log(x/sin(x)) = Sum_{n>0} (2^(2*n-1)*(-1)^(n+1)*B(2*n)/(n*(2*n)!) * x^(2*n)). - Ralf Stephan, Apr 01 2015 [corrected by Roland J. Etienne, Apr 19 2016]

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).

A340472 Numerators of an approximation to zeta(n)/Pi^n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 61, 1, 277, 1, 50521, 691, 540553, 2, 199360981, 3617, 3878302429, 43867, 2404879675441, 174611, 14814847529501, 155366, 69348874393137901, 236364091, 238685140977801337, 1315862, 4087072509293123892361, 6785560294, 13181680435827682794403, 6892673020804

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. A046988, A340471 (denominators).

a[k_] := Numerator[(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))); numerator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021

a(n) = numerator 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) = A046988(n).

A351864 Numerator of zeta({6}_n)/Pi^(6n).

Original entry on oeis.org

1, 1, 4, 2, 4, 1, 4, 4, 4, 4, 16, 2, 4, 2, 8, 8, 4, 4, 16, 8, 16, 1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 2, 4, 2, 8, 8, 4, 4, 16, 8, 16, 2, 8, 8, 8, 8, 32, 8, 16, 8, 32, 32, 4, 4, 16, 8, 16, 4, 16

Offset: 0

Roudy El Haddad, Feb 22 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. A351806 (denominators).

Cf. A000120, A240883, A046988, A013664, A103345.

a[n_] := Numerator[6*2^(6*n)/(6*n + 3)!]; Array[a, 71, 0]

PARI
```
a(n) = 1 << (hammingweight(3*n+1) - 1);
```

a(n) = numerator(6*2^(6*n)/(6*n + 3)!).
a(n) = 2^(A000120(3*n + 1) - 1).
a(n) = 2^A240883(n).

A188060 Numerator of 8^(2n-1) |B(2n)| / (2n)!, where B() are the Bernoulli numbers.

Original entry on oeis.org

8, 128, 4096, 32768, 1048576, 11593056256, 536870912, 7767448354816, 3014517285978112, 191986824837595136, 2733227576976736256, 66530577009460375453696, 5926115612870995607552, 488951148984934932554973184, 7946710949908368748447692488704, 71105936114697022329949662478336

Offset: 1

Eric Desbiaux, Apr 15 2011

nonn

Start with zeta(2n) = (2Pi)^(2n) |B(2n)| /(2 (2n)!) and replace Pi by 4 arctan(1) and take the numerator of the rational part. The denominator is given by A036278.

			8/3,128/45,4096/945,32768/4725,...

Cf. A002432, A046988, A036278.

 f:=n->8^(2*n)*abs(B(2*n))/(2*(2*n)!); [seq(numer(f(n)),n=1..60)];

Entry revised by N. J. A. Sloane, Apr 17 2011

cp's OEIS Frontend

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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A231327 Denominator of rational component of zeta(4n)/zeta(2n).

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A283301 Numerators of coefficients at even powers in Taylor series expansion of log(x/sin(x)).

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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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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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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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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

Sage

Formula

A340472 Numerators of an approximation to zeta(n)/Pi^n.

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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/(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).